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<div class="section" id="fwk-redden-ch04_s06" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">4.6</span> Rational Functions: Addition and Subtraction</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch04_s06_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch04_s06_o01" numeration="arabic">
<li>Add and subtract rational functions.</li>
<li>Simplify complex rational expressions.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch04_s06_s01" version="5.0" lang="en">
<h2 class="title editable block">Adding and Subtracting Rational Functions</h2>
<p class="para block" id="fwk-redden-ch04_s06_s01_p01">Adding and subtracting rational expressions is similar to adding and subtracting fractions. Recall that if the denominators are the same, we can add or subtract the numerators and write the result over the common denominator. When working with rational expressions, the common denominator will be a polynomial. In general, given polynomials <em class="emphasis">P</em>, <em class="emphasis">Q</em>, and <em class="emphasis">R</em>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1798" display="inline"><mrow><mi>Q</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, we have the following:</p>
<p class="para block" id="fwk-redden-ch04_s06_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1799" display="block"><mrow><mfrac><mi>P</mi><mi>Q</mi></mfrac><mo>±</mo><mfrac><mi>R</mi><mi>Q</mi></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mo>±</mo><mi>R</mi></mrow><mi>Q</mi></mfrac></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s06_s01_p03">The set of restrictions to the domain of a sum or difference of rational expressions consists of the restrictions to the domains of each expression.</p>
<div class="callout block" id="fwk-redden-ch04_s06_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch04_s06_s01_p04">Subtract: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1800" display="inline"><mtext> </mtext><mrow><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p05">The denominators are the same. Hence we can subtract the numerators and write the result over the common denominator. Take care to distribute the negative 1.</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p06"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1801" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn></mrow></mfrac></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn></mrow></mfrac><mstyle color="#007fbf"><mtext> </mtext><mi>S</mi><mi>u</mi><mi>b</mi><mi>t</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>n</mi><mi>u</mi><mi>m</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>s</mi><mtext>.</mtext></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn></mrow></mfrac><mtext> </mtext><mstyle color="#007fbf"><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>f</mi><mi>y</mi><mtext>.</mtext></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mover><mrow><menclose notation="updiagonalstrike"><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow></menclose></mrow><mn>1</mn></mover></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>C</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>l</mi><mo>.</mo></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mi>R</mi><mi>e</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mtext> </mtext></mstyle><mstyle color="#007fbf"><mi>x</mi><mo>≠</mo><mo>±</mo><mn>8</mn></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p07">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1802" display="inline"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow></mfrac></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1803" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>±</mo><mn>8</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s06_s01_p08">To add rational expressions with unlike denominators, first find equivalent expressions with common denominators. Do this just as you have with fractions. If the denominators of fractions are relatively prime, then the least common denominator (LCD) is their product. For example,</p>
<p class="para block" id="fwk-redden-ch04_s06_s01_p09"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1804" display="block"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>y</mi></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>⇒</mo></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>LCD</mtext><mo>=</mo><mi>x</mi><mo>⋅</mo><mi>y</mi><mo>=</mo><mi>x</mi><mi>y</mi></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s06_s01_p10">Multiply each fraction by the appropriate form of 1 to obtain equivalent fractions with a common denominator.</p>
<p class="para block" id="fwk-redden-ch04_s06_s01_p11"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1805" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>y</mi></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>1</mn><mstyle color="#007fbf"><mrow><mo>⋅</mo><mi>y</mi></mrow></mstyle></mrow><mrow><mi>x</mi><mstyle color="#007fbf"><mrow><mo>⋅</mo><mi>y</mi></mrow></mstyle></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn><mstyle color="#007fbf"><mrow><mo>⋅</mo><mi>x</mi></mrow></mstyle></mrow><mrow><mi>y</mi><mstyle color="#007fbf"><mrow><mo>⋅</mo><mi>x</mi></mrow></mstyle></mrow></mfrac><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mi>y</mi><mrow><mi>x</mi><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>E</mi><mi>q</mi><mi>u</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>e</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mtext> </mtext><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>a</mi><mtext> </mtext><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s06_s01_p12">In general, given polynomials <em class="emphasis">P</em>, <em class="emphasis">Q</em>, <em class="emphasis">R</em>, and <em class="emphasis">S</em>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1806" display="inline"><mrow><mi>Q</mi><mo>≠</mo><mn>0</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1807" display="inline"><mrow><mi>S</mi><mo>≠</mo><mn>0</mn></mrow></math></span>, we have the following:</p>
<p class="para block" id="fwk-redden-ch04_s06_s01_p13"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1808" display="block"><mrow><mfrac><mi>P</mi><mi>Q</mi></mfrac><mo>±</mo><mfrac><mi>R</mi><mi>S</mi></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mi>S</mi><mo>±</mo><mi>Q</mi><mi>R</mi></mrow><mrow><mi>Q</mi><mi>S</mi></mrow></mfrac></mrow></math></span></p>
<div class="callout block" id="fwk-redden-ch04_s06_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch04_s06_s01_p14">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1809" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>5</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1810" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1811" display="inline"><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow></math></span> and state the restrictions.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p15">Here the LCD is the product of the denominators <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1812" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Multiply by the appropriate factors to obtain rational expressions with a common denominator before adding.</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p16"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1813" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>+</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>5</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>5</mn><mi>x</mi></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle><mo>+</mo><mfrac><mn>2</mn><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>5</mn><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>5</mn><mi>x</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>11</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p17">The domain of <em class="emphasis">f</em> consists all real numbers except <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1814" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span>, and the domain of <em class="emphasis">g</em> consists of all real numbers except −1. Therefore, the domain of <em class="emphasis">f</em> + <em class="emphasis">g</em> consists of all real numbers except −1 and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1815" display="inline"><mrow><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p18">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1816" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1817" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s06_s01_p19">It is not always the case that the LCD is the product of the given denominators. Typically, the denominators are not relatively prime; thus determining the LCD requires some thought. Begin by factoring all denominators. The LCD is the product of all factors with the highest power.</p>
<div class="callout block" id="fwk-redden-ch04_s06_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch04_s06_s01_p20">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1818" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1819" display="inline"><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mn>14</mn><mi>x</mi></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span>, find <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1820" display="inline"><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow></math></span> and state the restrictions to the domain.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p21">To determine the LCD, factor the denominator of <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1821" display="inline"><mi>g</mi><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p22"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1822" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo>)</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mn>14</mn><mi>x</mi></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mn>14</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p23">In this case the <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1823" display="inline"><mrow><mtext>LCD</mtext><mo>=</mo><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Multiply <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1824" display="inline"><mi>f</mi><mtext> </mtext></math></span> by 1 in the form of <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1825" display="inline"><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></math></span> to obtain equivalent algebraic fractions with a common denominator and then subtract.</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p24"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1826" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mrow><mstyle color="#007fbf"><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mstyle></mrow></mrow><mrow><mstyle color="#007fbf"><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mstyle></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mn>14</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>3</mn><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>4</mn><mo>+</mo><mn>14</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>11</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p25">The domain of <em class="emphasis">f</em> consists of all real numbers except <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1827" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></math></span>, and the domain of <em class="emphasis">g</em> consists of all real numbers except 1 and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1828" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>.</mo></math></span> Therefore, the domain of <em class="emphasis">f</em> − <em class="emphasis">g</em> consists of all real numbers except 1 and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1829" display="inline"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p26">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1830" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1831" display="inline"><mrow><mi>x</mi><mo>≠</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mn>1</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s06_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch04_s06_s01_p27">Simplify and state the restrictions: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1832" display="inline"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>6</mn><mo>−</mo><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>18</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>36</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p28">Begin by applying the opposite binomial property <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1833" display="inline"><mrow><mn>6</mn><mo>−</mo><mi>x</mi><mo>=</mo><mo>−</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1834" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>6</mn><mo>−</mo><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>18</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>36</mn></mrow></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="left"><mo>=</mo><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mstyle color="#007fbf"><mrow><mo>−</mo><mn>1</mn><mo>⋅</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mfrac><mo>−</mo><mfrac><mrow><mn>18</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd columnalign="left"><mo>=</mo><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>18</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr></mtable><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p30">Next, find equivalent fractions with the <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1835" display="inline"><mrow><mi>L</mi><mi>C</mi><mi>D</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span> and then simplify.</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p31"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1836" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mstyle><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mstyle><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mstyle><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow></mstyle><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>18</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>18</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>18</mn><mi>x</mi><mo>−</mo><mn>18</mn><mi>x</mi><mo>+</mo><mn>36</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>36</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p32">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1837" display="inline"><mtext> </mtext><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mfrac></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1838" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>±</mo><mn>6</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s06_s01_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s06_s01_p33"><strong class="emphasis bold">Try this!</strong> Simplify and state the restrictions: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1839" display="inline"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>4</mn><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p34">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1840" display="inline"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1841" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>±</mo><mn>1</mn></mrow></math></span></p>
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<p class="para editable block" id="fwk-redden-ch04_s06_s01_p36">Rational expressions are sometimes expressed using negative exponents. In this case, apply the rules for negative exponents before simplifying the expression.</p>
<div class="callout block" id="fwk-redden-ch04_s06_s01_n05">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch04_s06_s01_p37">Simplify and state the restrictions: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1842" display="inline"><mrow><mn>5</mn><msup><mi>a</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p38">Recall that <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1843" display="inline"><mrow><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac></mrow></math></span>. Begin by rewriting the rational expressions with negative exponents as fractions.</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p39"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1844" display="block"><mrow><mn>5</mn><msup><mi>a</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>5</mn><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>1</mn></msup></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p40">Then find the LCD and add.</p>
<p class="para" id="fwk-redden-ch04_s06_s01_p41"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1845" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mn>5</mn><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>1</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>5</mn><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow><mrow><mstyle color="#007fbf"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mstyle color="#007fbf"><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mstyle></mrow><mrow><mstyle color="#007fbf"><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></mstyle></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>5</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>E</mi><mi>q</mi><mi>u</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>e</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>r</mi><mi>e</mi><mi>s</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mtext> </mtext><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>a</mi><mtext> </mtext><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>10</mn><mi>a</mi><mo>+</mo><mn>25</mn><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>d</mi><mi>d</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>a</mi><mo>+</mo><mn>25</mn></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mrow><mi>S</mi><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>f</mi><mi>i</mi><mi>y</mi><mtext>.</mtext></mrow></mstyle></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s01_p42">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1846" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>a</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1847" display="inline"><mrow><mi>a</mi><mo>≠</mo><mo>−</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mo>,</mo><mn>0</mn></mrow></math></span></p>
</div>
</div>
<div class="section" id="fwk-redden-ch04_s06_s02" version="5.0" lang="en">
<h2 class="title editable block">Simplifying Complex Rational Expressions</h2>
<p class="para block" id="fwk-redden-ch04_s06_s02_p01">A <span class="margin_term"><a class="glossterm">complex rational expression</a><span class="glossdef">A rational expression that contains one or more rational expressions in the numerator or denominator or both.</span></span> is defined as a rational expression that contains one or more rational expressions in the numerator or denominator or both. For example,
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1848" display="block"><mrow><mfrac><mrow><mtext> </mtext><mn>4</mn><mo>−</mo><mfrac><mn>12</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext></mrow><mrow><mn>2</mn><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
is a complex rational expression. We simplify a complex rational expression by finding an equivalent fraction where the numerator and denominator are polynomials. There are two methods for simplifying complex rational expressions, and we will outline the steps for both methods. For the sake of clarity, assume that variable expressions used as denominators are nonzero.</p>
<div class="section" id="fwk-redden-ch04_s06_s02_s01" version="5.0" lang="en">
<h2 class="title editable block">Method 1: Simplify Using Division</h2>
<p class="para editable block" id="fwk-redden-ch04_s06_s02_s01_p01">We begin our discussion on simplifying complex rational expressions using division. Before we can multiply by the reciprocal of the denominator, we must simplify the numerator and denominator separately. The goal is to first obtain single algebraic fractions in the numerator and the denominator. The steps for simplifying a complex algebraic fraction are illustrated in the following example.</p>
<div class="callout block" id="fwk-redden-ch04_s06_s02_s01_n01">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p02">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1849" display="inline"><mrow><mfrac><mrow><mtext> </mtext><mn>4</mn><mo>−</mo><mfrac><mn>12</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext></mrow><mrow><mn>2</mn><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p03"><strong class="emphasis bold">Step 1:</strong> Simplify the numerator and denominator to obtain a single algebraic fraction divided by another single algebraic fraction. In this example, find equivalent terms with a common denominator in both the numerator and denominator before adding and subtracting.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1850" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mtext> </mtext><mn>4</mn><mo>−</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext></mrow><mrow><mn>2</mn><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mfrac><mn>4</mn><mn>1</mn></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mstyle><mo>−</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mi>x</mi><mi>x</mi></mfrac></mrow></mstyle><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mfrac><mn>2</mn><mn>1</mn></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mstyle><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mi>x</mi><mi>x</mi></mfrac></mrow></mstyle><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>12</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext><mtext> </mtext></mrow><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>5</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>E</mi><mi>q</mi><mi>u</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>e</mi><mi>n</mi><mi>t</mi><mtext> </mtext><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mtext> </mtext><mi>w</mi><mi>i</mi><mi>t</mi><mi>h</mi><mtext> </mtext><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>o</mi><mi>n</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>s</mi><mtext>.</mtext></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext><mtext> </mtext></mrow><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>A</mi><mi>d</mi><mi>d</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mtext> </mtext><mi>i</mi><mi>n</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>n</mi><mi>u</mi><mi>m</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext>.</mtext></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p05">At this point we have a single algebraic fraction divided by another single algebraic fraction.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p06"><strong class="emphasis bold">Step 2:</strong> Multiply the numerator by the reciprocal of the denominator.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p07"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1851" display="block"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext><mtext> </mtext></mrow><mrow><mstyle color="#007fbf"><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mstyle></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></mstyle></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p08"><strong class="emphasis bold">Step 3:</strong> Factor all numerators and denominators completely.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p09"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1852" display="block"><mrow><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p10"><strong class="emphasis bold">Step 4:</strong> Cancel all common factors.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p11"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1853" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><menclose notation="updiagonalstrike"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></menclose></mrow></mfrac><mo>⋅</mo><mfrac><mrow><menclose notation="updiagonalstrike"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></menclose></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p12">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1854" display="inline"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s06_s02_s01_n02">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p13">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1855" display="inline"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>7</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext></mrow><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>5</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p14">Obtain a single algebraic fraction in the numerator and in the denominator.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p15"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1856" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>7</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mtext> </mtext><mtext> </mtext></mrow><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>5</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow></mfrac></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mstyle><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mstyle><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mn>7</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mstyle><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mstyle><mo>)</mo></mrow></mrow></mfrac></mrow><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mstyle><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mstyle><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mn>5</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mstyle><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mstyle color="#007fbf"><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mstyle><mo>)</mo></mrow></mrow></mfrac></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>2</mn><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>7</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mtext> </mtext><mtext> </mtext></mrow><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>5</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>7</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mtext> </mtext><mtext> </mtext></mrow><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>13</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p16">Next, multiply the numerator by the reciprocal of the denominator, factor, and then cancel.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p17"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1857" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>13</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>⋅</mo><mfrac><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p18">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1858" display="inline"><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span></p>
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<div class="callout block" id="fwk-redden-ch04_s06_s02_s01_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p19"><strong class="emphasis bold">Try this!</strong> Simplify using division: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1859" display="inline"><mrow><mfrac><mrow><mtext> </mtext><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext></mrow><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfrac></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p20">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1860" display="inline"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></mrow></math></span></p>
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<p class="para editable block" id="fwk-redden-ch04_s06_s02_s01_p22">Sometimes complex rational expressions are expressed using negative exponents.</p>
<div class="callout block" id="fwk-redden-ch04_s06_s02_s01_n03">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p23">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1861" display="inline"><mrow><mfrac><mrow><mn>2</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>4</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p24">We begin by rewriting the expression without negative exponents.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p25"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1862" display="block"><mrow><mfrac><mrow><mn>2</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>4</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mfrac><mn>2</mn><mi>y</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>4</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p26">Obtain single algebraic fractions in the numerator and denominator and then multiply by the reciprocal of the denominator.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p27"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1863" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mfrac><mrow><mfrac><mn>2</mn><mi>y</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>4</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></mrow><mrow><mtext> </mtext><mtext> </mtext><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext><mtext> </mtext></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p28">Apply the opposite binomial property <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1864" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> and then cancel.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1865" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></menclose></mrow><mrow><menclose notation="updiagonalstrike"><mi>x</mi></menclose><menclose notation="updiagonalstrike"><mi>y</mi></menclose></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mover><mrow><menclose notation="updiagonalstrike"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></menclose></mrow><mi>x</mi></mover><mover><mrow><menclose notation="updiagonalstrike"><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></menclose></mrow><mi>y</mi></mover></mrow><mrow><mo>−</mo><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></menclose><mrow><mo>(</mo><mrow><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s01_p30">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1866" display="inline"><mrow><mo>−</mo><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfrac></mrow></math></span></p>
</div>
</div>
<div class="section" id="fwk-redden-ch04_s06_s02_s02" version="5.0" lang="en">
<h2 class="title editable block">Method 2: Simplify Using the LCD</h2>
<p class="para editable block" id="fwk-redden-ch04_s06_s02_s02_p01">An alternative method for simplifying complex rational expressions involves clearing the fractions by multiplying the expression by a special form of 1. In this method, multiply the numerator and denominator by the least common denominator (LCD) of all given fractions.</p>
<div class="callout block" id="fwk-redden-ch04_s06_s02_s02_n01">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p02">Simplify: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1867" display="inline"><mrow><mfrac><mrow><mtext> </mtext><mn>4</mn><mo>−</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext></mrow><mrow><mn>2</mn><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p03"><strong class="emphasis bold">Step 1:</strong> Determine the LCD of all the fractions in the numerator and denominator. In this case, the denominators of the given fractions are 1, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1868" display="inline"><mi>x</mi></math></span>, and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1869" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span> Therefore, the LCD is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1870" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p04"><strong class="emphasis bold">Step 2:</strong> Multiply the numerator and denominator by the LCD. This step should clear the fractions in both the numerator and denominator.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p05"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1871" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mfrac><mrow><mtext> </mtext><mn>4</mn><mo>−</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext></mrow><mrow><mn>2</mn><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mo>−</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo>⋅</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle></mrow><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow><mo>⋅</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>M</mi><mi>u</mi><mi>l</mi><mi>t</mi><mi>i</mi><mi>p</mi><mi>l</mi><mi>y</mi><mtext> </mtext><mi>n</mi><mi>u</mi><mi>m</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>n</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mtext> </mtext><mi>L</mi><mi>C</mi><mi>D</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mtext> </mtext><mn>4</mn><mo>⋅</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>−</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>+</mo><mfrac><mn>9</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mtext> </mtext></mrow><mrow><mtext> </mtext><mn>2</mn><mo>⋅</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle><mo>+</mo><mfrac><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>⋅</mo><mstyle color="#007fbf"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mstyle></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mtext> </mtext><mi>D</mi><mi>i</mi><mi>s</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>t</mi><mi>h</mi><mi>e</mi><mi>n</mi><mtext> </mtext><mi>c</mi><mi>a</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>l</mi><mo>.</mo></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable><mtext> </mtext></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p06">This leaves us with a single algebraic fraction with a polynomial in the numerator and in the denominator.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p07"><strong class="emphasis bold">Step 3:</strong> Factor the numerator and denominator completely.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1872" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mtext> </mtext></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p09"><strong class="emphasis bold">Step 4:</strong> Cancel all common factors.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch04_m1873" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><menclose notation="updiagonalstrike"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></menclose></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p11"><strong class="emphasis bold">Note</strong>: This was the same problem presented in Example 6 and the results here are the same. It is worth taking the time to compare the steps involved using both methods on the same problem.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p12">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1874" display="inline"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s06_s02_s02_p13">It is important to point out that multiplying the numerator and denominator by the same nonzero factor is equivalent to multiplying by 1 and does not change the problem.</p>
<div class="callout block" id="fwk-redden-ch04_s06_s02_s02_n01a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p14"><strong class="emphasis bold">Try this!</strong> Simplify using the LCD: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1875" display="inline"><mrow><mfrac><mrow><mtext> </mtext><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mtext> </mtext></mrow><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfrac></mrow></math></span>.</p>
<p class="para" id="fwk-redden-ch04_s06_s02_s02_p15">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1876" display="inline"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></mrow></math></span></p>
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</div>
</div>
<div class="key_takeaways editable block" id="fwk-redden-ch04_s06_s02_s02_n02">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch04_s06_s02_s02_l01" mark="bullet">
<li>Adding and subtracting rational expressions is similar to adding and subtracting fractions. A common denominator is required. If the denominators are the same, then we can add or subtract the numerators and write the result over the common denominator.</li>
<li>The set of restrictions to the domain of a sum or difference of rational functions consists of the restrictions to the domains of each function.</li>
<li>Complex rational expressions can be simplified into equivalent expressions with a polynomial numerator and polynomial denominator. They are reduced to lowest terms if the numerator and denominator are polynomials that share no common factors other than 1.</li>
<li>One method of simplifying a complex rational expression requires us to first write the numerator and denominator as a single algebraic fraction. Then multiply the numerator by the reciprocal of the denominator and simplify the result.</li>
<li>Another method for simplifying a complex rational expression requires that we multiply it by a special form of 1. Multiply the numerator and denominator by the LCD of all the denominators as a means to clear the fractions. After doing this, simplify the remaining rational expression.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch04_s06_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd01">
<h3 class="title">Part A: Adding and Subtracting Rational Functions</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch04_s06_qs01_p01"><strong class="emphasis bold">State the restrictions and simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa01">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1877" display="block"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa02">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1880" display="block"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa03">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1883" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa04">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1886" display="block"><mrow><mfrac><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa05">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1889" display="block"><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>−</mo><mn>2</mn><mi>x</mi></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa06">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1892" display="block"><mrow><mfrac><mn>4</mn><mrow><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa07">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1895" display="block"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mn>5</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa08">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1898" display="block"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>7</mn></mrow></mfrac><mo>−</mo><mn>1</mn></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa09">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1901" display="block"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa10">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1904" display="block"><mrow><mfrac><mn>2</mn><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa11">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1907" display="block"><mrow><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa12">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1910" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa13">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1913" display="block"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>7</mn><mo>−</mo><mi>x</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa14">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1916" display="block"><mrow><mfrac><mn>2</mn><mrow><mn>8</mn><mo>−</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa15">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1919" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></mfrac><mi> </mi><mo>−</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mfrac><mi> </mi></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa16">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1922" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mi> </mi><mo>−</mo><mfrac><mi>x</mi><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa17">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1925" display="block"><mrow><mfrac><mi>x</mi><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa18">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1928" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>−</mo><mfrac><mn>3</mn><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa19">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1931" display="block"><mrow><mfrac><mrow><mn>5</mn><mo>−</mo><mi>x</mi></mrow><mrow><mn>7</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>49</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa20">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1934" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa21">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1937" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa22">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1940" display="block"><mrow><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mi>x</mi></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa23">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1943" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa24">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1946" display="block"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>6</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa25">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1949" display="block"><mrow><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn></mrow><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa26">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1952" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd01_qd02" start="27">
<p class="para" id="fwk-redden-ch04_s06_qs01_p54"><strong class="emphasis bold">Given</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1955" display="inline"><mi>f</mi></math></span> <strong class="emphasis bold">and</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1956" display="inline"><mi>g</mi></math></span><strong class="emphasis bold">, simplify the sum</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1957" display="inline"><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow></math></span> <strong class="emphasis bold">and difference</strong> <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1958" display="inline"><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo>.</mo></math></span> <strong class="emphasis bold">Also, state the domain using interval notation.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa27">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1959" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>5</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa28">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1963" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa29">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1967" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa30">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1971" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mi>x</mi><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa31">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1975" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>6</mn><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>18</mn></mrow><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa32">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1979" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa33">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1983" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mi>x</mi><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa34">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1987" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mi>x</mi><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa35">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1991" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa36">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1995" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>6</mn><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>13</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac></mrow><mtext>, </mtext><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mn>2</mn><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>10</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd01_qd03" start="37">
<p class="para" id="fwk-redden-ch04_s06_qs01_p75"><strong class="emphasis bold">State the restrictions and simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa37">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m1999" display="block"><mrow><mn>1</mn><mo>+</mo><mfrac><mn>3</mn><mi>x</mi></mfrac><mo>−</mo><mfrac><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa38">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2002" display="block"><mrow><mn>4</mn><mo>+</mo><mfrac><mn>2</mn><mi>x</mi></mfrac><mo>−</mo><mfrac><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa39">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2005" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>23</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa40">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2008" display="block"><mrow><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>10</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>19</mn><mi>x</mi><mo>+</mo><mn>18</mn></mrow><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa41">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2011" display="block"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa42">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2014" display="block"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa43">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2017" display="block"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa44">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2020" display="block"><mrow><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>4</mn><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa45">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2023" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn><mi>x</mi><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>2</mn><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa46">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2025" display="block"><mrow><mfrac><mrow><mn>10</mn><mi>x</mi></mrow><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>5</mn><mi>x</mi></mrow><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd01_qd04" start="47">
<p class="para" id="fwk-redden-ch04_s06_qs01_p96"><strong class="emphasis bold">Simplify the given algebraic expressions. Assume all variable expressions in the denominator are nonzero.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2027" display="inline"><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2029" display="inline"><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2031" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2033" display="inline"><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>4</mn><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2035" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2037" display="inline"><mrow><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mi>y</mi><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2039" display="inline"><mrow><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2041" display="inline"><mrow><mn>2</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2043" display="inline"><mrow><msup><mi>a</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2045" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2047" display="inline"><mrow><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2049" display="inline"><mrow><mi>x</mi><msup><mi>y</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup><mo>+</mo><mi>y</mi><msup><mi>x</mi><mrow><mo>−</mo><mi>n</mi></mrow></msup></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd02">
<h3 class="title">Part B: Simplifying Complex Rational Expressions</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd02_qd01" start="59">
<p class="para" id="fwk-redden-ch04_s06_qs01_p121"><strong class="emphasis bold">Simplify. Assume all variable expressions in the denominators are nonzero.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa59">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2051" display="block"><mrow><mfrac><mrow><mfrac><mrow><mn>75</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mfrac><mrow><mn>25</mn><msup><mi>x</mi><mn>3</mn></msup></mrow><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa60">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2053" display="block"><mrow><mfrac><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mrow><mn>36</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></mrow><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa61">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2055" display="block"><mrow><mfrac><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>36</mn></mrow><mrow><mn>32</mn><msup><mi>x</mi><mn>5</mn></msup></mrow></mfrac></mrow><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa62">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2057" display="block"><mrow><mfrac><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mrow><mn>56</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>64</mn></mrow><mrow><mn>7</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa63">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2059" display="block"><mrow><mfrac><mrow><mfrac><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>10</mn></mrow></mfrac></mrow><mrow><mfrac><mrow><mn>25</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa64">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2061" display="block"><mrow><mfrac><mrow><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>27</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa65">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2063" display="block"><mrow><mfrac><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mrow><mtext> </mtext><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mtext> </mtext></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa66">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2065" display="block"><mrow><mfrac><mrow><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mrow><mrow><mtext> </mtext><mfrac><mrow><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mtext> </mtext></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa67">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2067" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>−</mo><mfrac><mn>3</mn><mi>x</mi></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa68">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2069" display="block"><mrow><mfrac><mrow><mfrac><mn>4</mn><mi>x</mi></mfrac><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa69">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2071" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mn>9</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa70">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2073" display="block"><mrow><mfrac><mrow><mfrac><mn>2</mn><mn>5</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mrow><mfrac><mn>4</mn><mrow><mn>25</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa71">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2075" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mn>36</mn></mrow><mrow><mn>6</mn><mo>−</mo><mfrac><mn>1</mn><mi>y</mi></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa72">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2077" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mn>5</mn></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>y</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>25</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa73">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2079" display="block"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mfrac><mn>6</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>8</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mn>3</mn><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mo>−</mo><mfrac><mn>2</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa74">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2081" display="block"><mrow><mfrac><mrow><mn>2</mn><mo>+</mo><mfrac><mrow><mn>13</mn></mrow><mi>x</mi></mfrac><mo>−</mo><mfrac><mn>7</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mn>3</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>−</mo><mfrac><mrow><mn>10</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa75">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2083" display="block"><mrow><mfrac><mrow><mn>9</mn><mo>−</mo><mfrac><mrow><mn>12</mn></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mn>9</mn><mo>−</mo><mfrac><mn>4</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa76">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2085" display="block"><mrow><mfrac><mrow><mn>4</mn><mo>−</mo><mfrac><mrow><mn>25</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mn>4</mn><mo>−</mo><mfrac><mn>8</mn><mi>x</mi></mfrac><mo>−</mo><mfrac><mn>5</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa77">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2087" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>5</mn><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mrow><mfrac><mn>2</mn><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa78">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2089" display="block"><mrow><mfrac><mrow><mfrac><mn>2</mn><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>−</mo><mfrac><mn>3</mn><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa79">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2091" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow><mrow><mfrac><mn>2</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa80">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2093" display="block"><mrow><mfrac><mrow><mfrac><mn>4</mn><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac></mrow><mrow><mfrac><mn>3</mn><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa81">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2095" display="block"><mrow><mfrac><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mi> </mi><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mi> </mi></mrow><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mi> </mi><mo>−</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mi> </mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa82">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2097" display="block"><mrow><mfrac><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac><mi> </mi><mo>−</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mi> </mi></mrow><mrow><mfrac><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mi> </mi><mi> </mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa83">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2099" display="block"><mrow><mfrac><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa84">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2101" display="block"><mrow><mfrac><mrow><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa85">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2102" display="block"><mrow><mfrac><mrow><mtext> </mtext><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></mfrac><mtext> </mtext></mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa86">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2104" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mrow><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>6</mn><mi>x</mi></mrow><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa87">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2106" display="block"><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa88">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2108" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa89">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2110" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa90">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2112" display="block"><mrow><mfrac><mrow><mfrac><mn>2</mn><mi>y</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow><mrow><mfrac><mn>4</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa91">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2114" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mrow><mn>25</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mi> </mi><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>5</mn><mi>y</mi></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa92">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2116" display="block"><mrow><mfrac><mrow><mn>16</mn><msup><mi>y</mi><mn>2</mn></msup><mi> </mi><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mi> </mi><mo>−</mo><mn>4</mn><mi>y</mi></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa93">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2118" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mi>b</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>a</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>b</mi><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa94">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2120" display="block"><mrow><mfrac><mrow><mfrac><mn>1</mn><mi>a</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>b</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>b</mi><mn>3</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><msup><mi>a</mi><mn>3</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa95">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2122" display="block"><mrow><mfrac><mrow><mfrac><mi>x</mi><mi>y</mi></mfrac><mo>−</mo><mfrac><mi>y</mi><mi>x</mi></mfrac></mrow><mrow><mfrac><mn>1</mn><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mi>y</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa96">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2124" display="block"><mrow><mfrac><mrow><mtext> </mtext><mtext> </mtext><mfrac><mn>2</mn><mi>y</mi></mfrac><mo>−</mo><mfrac><mn>5</mn><mi>x</mi></mfrac><mtext> </mtext><mtext> </mtext></mrow><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mfrac><mrow><mn>25</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mi>x</mi></mfrac></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa97">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2126" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa98">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2128" display="block"><mrow><mfrac><mrow><msup><mi>y</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mn>25</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mrow><mn>5</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mi>y</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa99">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2130" display="block"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><mi>x</mi><mo>−</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa100">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2132" display="block"><mrow><mfrac><mrow><mn>16</mn><mi> </mi><mo>−</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi> </mi><mo>−</mo><mn>4</mn></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa101">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2134" display="block"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mn>4</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>21</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mrow><mn>1</mn><mo>−</mo><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>15</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa102">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2136" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>4</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><mn>3</mn><mo>−</mo><mn>8</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>16</mn><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa103">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2138" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>2</mn><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>3</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa104">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2140" display="block"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow><mrow><msup><mi>x</mi><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa105">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p214">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2142" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span>, simplify <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2143" display="inline"><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa106">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p216">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2145" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>, simplify <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2146" display="inline"><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mi>b</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></mrow><mrow><mi>b</mi><mo>−</mo><mi>a</mi></mrow></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa107">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p218">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2148" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></math></span>, simplify the difference quotient <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2149" display="inline"><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>h</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mi>h</mi></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa108">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p220">Given <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2151" display="inline"><mtext> </mtext><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mn>1</mn></mrow></math></span>, simplify the difference quotient <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2152" display="inline"><mrow><mfrac><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>h</mi></mrow><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mi>h</mi></mfrac></mrow><mo>.</mo></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd03">
<h3 class="title">Part C: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s06_qs01_qd03_qd01" start="109">
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa109">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p222">Explain why the domain of a sum of rational functions is the same as the domain of the difference of those functions.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa110">
<div class="question">
<p class="para" id="fwk-redden-ch04_s06_qs01_p223">Two methods for simplifying complex rational expressions have been presented in this section. Which of the two methods do you feel is more efficient, and why?</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch04_s06_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p03_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1878" display="inline"><mrow><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1879" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>−</mo><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa02_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p07_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1884" display="inline"><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1885" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa04_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p11_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1890" display="inline"><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mi>x</mi></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1891" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa06_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p15_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1896" display="inline"><mrow><mfrac><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1897" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa08_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p19_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1902" display="inline"><mrow><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1903" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>−</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mo>,</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa10_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p23_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1908" display="inline"><mrow><mfrac><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1909" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa12_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p27_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1914" display="inline"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1915" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa14_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p31_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1920" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1921" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>±</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa16_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p35_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1926" display="inline"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1927" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa18_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p39_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1932" display="inline"><mrow><mfrac><mrow><mn>7</mn><mrow><mo>(</mo><mrow><mn>5</mn><mo>−</mo><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mn>7</mn><mo>+</mo><mi>x</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>7</mn><mo>−</mo><mi>x</mi></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1933" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>−</mo><mn>7</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa20_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa21_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p43_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1938" display="inline"><mrow><mfrac><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1939" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa22_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa23_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p47_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1944" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn></mrow><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1945" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa24_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa25_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p51_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1950" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1951" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>±</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa26_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa27_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p56_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1960" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1961" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>; Domain: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1962" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa28_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa29_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p60_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1968" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1969" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mrow><mn>8</mn><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; Domain: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1970" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa30_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa31_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p64_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1976" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>6</mn><mrow><mo>(</mo><mrow><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mrow><mi>x</mi><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1977" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>6</mn><mrow><mi>x</mi><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow></math></span>; Domain: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1978" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mn>0</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa32_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa33_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p68_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1984" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1985" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; Domain: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1986" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn><mo>,</mo><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>5</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa34_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa35_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p72_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1992" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1993" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mfrac><mrow><mn>7</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; Domain: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1994" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mi>∞</mi><mo>,</mo><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>∪</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>,</mo><mi>∞</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa36_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa37_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p77_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2000" display="inline"><mrow><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2001" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa38_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa39_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p81_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2006" display="inline"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>8</mn></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2007" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>−</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>,</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa40_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa41_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p85_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2012" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2013" display="inline"><mrow><mi>x</mi><mo>≠</mo><mo>±</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa42_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa43_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p89_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m2018" display="inline"><mrow><mfrac><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mi>x</mi></mfrac></mrow></math></span>; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2019" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa44_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa45_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s06_qs01_p93_ans">0; <span class="inlineequation"><math xml:id="fwk-redden-ch04_m2024" display="inline"><mrow><mi>x</mi><mo>≠</mo><mn>0</mn><mo>,</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa46_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa47_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2028" display="block"><mrow><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa48_ans" audience="instructoronly">
<div class="answer" audience="instructoronly" d="" html="http://www.w3.org/1999/xhtml" mml="http://www.w3.org/1998/Math/MathML" xlink="http://www.w3.org/1999/xlink" xml="http://www.w3.org/XML/1998/namespace">
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s06_qs01_qa49_ans">
<div class="answer">
<span class="informalequation"><math xml:id="fwk-redden-ch04_m2032" display="block"><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mrow><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow></math></span>