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<title>Factoring Trinomials</title>
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<div class="section" id="fwk-redden-ch04_s03" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">4.3</span> Factoring Trinomials</h2>
<div class="learning_objectives block" id="fwk-redden-ch04_s03_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch04_s03_o01" numeration="arabic">
<li>Factor trinomials of the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0868" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow><mo>.</mo></math></span>
</li>
<li>Factor trinomials of higher degree.</li>
<li>Factor trinomials of the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0869" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow><mo>.</mo></math></span>
</li>
<li>Factor trinomials using the AC method.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch04_s03_s01" version="5.0" lang="en">
<h2 class="title block">Factoring Trinomials of the Form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0870" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span>
</h2>
<p class="para block" id="fwk-redden-ch04_s03_s01_p01">Some trinomials of the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0871" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> can be factored as a product of binomials. If a trinomial of this type factors, then we have:</p>
<p class="para block" id="fwk-redden-ch04_s03_s01_p02"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0872" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>m</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>n</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mi>x</mi><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>m</mi><mi>n</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>m</mi><mo stretchy="false">)</mo><mi>x</mi><mo>+</mo><mi>m</mi><mi>n</mi></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s03_s01_p03">This gives us</p>
<p class="para block" id="fwk-redden-ch04_s03_s01_p04"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0873" display="block"><mrow><mi>b</mi><mo>=</mo><mi>n</mi><mo>+</mo><mi>m</mi><mtext> and</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>c</mi><mo>=</mo><mi>m</mi><mi>n</mi></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s03_s01_p05">In short, if the leading coefficient of a factorable trinomial is 1, then the factors of the last term must add up to the coefficient of the middle term. This observation is the key to factoring trinomials using the technique known as the <span class="margin_term"><a class="glossterm">trial and error (or guess and check) method</a><span class="glossdef">Describes the method of factoring a trinomial by systematically checking factors to see if their product is the original trinomial.</span></span>.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch04_s03_s01_p06">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0874" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>20</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p07">We begin by writing two sets of blank parentheses. If a trinomial of this form factors, then it will factor into two linear binomial factors.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0875" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>20</mn><mo>=</mo><mrow><mo>(</mo><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p09">Write the factors of the first term in the first space of each set of parentheses. In this case, factor <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0876" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>x</mi><mo>⋅</mo><mi>x</mi></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p10"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0877" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>20</mn><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p11">Determine the factors of the last term whose sum equals the coefficient of the middle term. To do this, list all of the factorizations of 20 and search for factors whose sum equals 12.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p12"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0878" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>20</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>⋅</mo><mn>20</mn><mtext> </mtext></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>1</mn><mo>+</mo><mn>20</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>21</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>2</mn><mo>⋅</mo><mn>10</mn></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>2</mn><mo>+</mo><mn>10</mn></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>12</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mo>⋅</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>4</mn><mo>+</mo><mn>5</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p13">Choose 20 = 2 ⋅ 10 because 2 + 10 = 12. Write in the last term of each binomial using the factors determined in the previous step.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p14"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0879" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>20</mn><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mn>10</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p15">This can be visually interpreted as follows:</p>
<div class="informalfigure large">
<img src="section_07/58268cef0448982a94fc601452351d1b.png">
</div>
<p class="para" id="fwk-redden-ch04_s03_s01_p17">Check by multiplying the two binomials.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p18"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0880" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><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>10</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>20</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>20</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p19">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0881" display="inline"><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>10</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s03_s01_p20">Since multiplication is commutative, the order of the factors does not matter.</p>
<p class="para block" id="fwk-redden-ch04_s03_s01_p21"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0882" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>20</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>10</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s03_s01_p22">If the last term of the trinomial is positive, then either both of the constant factors must be negative or both must be positive.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch04_s03_s01_p23">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0883" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>12</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p24">First, factor <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0884" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>x</mi><mi>y</mi><mo>⋅</mo><mi>x</mi><mi>y</mi></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p25"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0885" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>12</mn><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p26">Next, search for factors of 12 whose sum is −7.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p27"><span class="informalequation">
<math xml:id="fwk-redden-ch04_m0886" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>12</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>⋅</mo><mn>12</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mo>−</mo><mn>1</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>13</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>⋅</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mo>−</mo><mn>2</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>8</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>3</mn><mo>⋅</mo><mn>4</mn></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mo>−</mo><mn>3</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mo>=</mo></mstyle></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>7</mn></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p28">In this case, choose −3 and −4 because <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0887" display="inline"><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>+</mo><mn>12</mn></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0888" display="inline"><mrow><mo>−</mo><mn>3</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>7</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p29"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0889" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>12</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p30">Check.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p31"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0890" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>12</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>12</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p32">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0891" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s03_s01_p33">If the last term of the trinomial is negative, then one of its factors must be negative.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s01_n03">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch04_s03_s01_p34">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0892" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p35">Begin by factoring the first term <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0893" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mi>x</mi><mo>⋅</mo><mi>x</mi></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p36"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0894" display="block"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p37">The factors of 12 are listed below. In this example, we are looking for factors whose sum is −4.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p38"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0895" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>12</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>⋅</mo><mn>12</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>1</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>11</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>2</mn><mo>⋅</mo><mn>6</mn><mtext> </mtext></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>2</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mo>=</mo></mstyle></mrow></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>4</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>3</mn><mo>⋅</mo><mn>4</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mo>→</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>3</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p39">Therefore, the coefficient of the last term can be factored as <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0896" display="inline"><mrow><mo>−</mo><mn>12</mn><mo>=</mo><mn>2</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0897" display="inline"><mrow><mn>2</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mn>4</mn></mrow><mo>.</mo></math></span> Because the last term has a variable factor of <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0898" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span>, use <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0899" display="inline"><mrow><mo>−</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mo>−</mo><mn>6</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> and factor the trinomial as follows:</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p40"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0900" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p41">Multiply to check.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p42"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0901" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn><mi>y</mi><mi>x</mi><mo>−</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p43">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0902" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s03_s01_p44">Often our first guess will not produce a correct factorization. This process may require repeated trials. For this reason, the check is very important and is not optional.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s01_n04">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch04_s03_s01_p45">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0903" display="inline"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>a</mi><mo>−</mo><mn>24</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p46">The first term of this trinomial, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0904" display="inline"><mrow><msup><mi>a</mi><mn>2</mn></msup></mrow></math></span>, factors as <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0905" display="inline"><mrow><mi>a</mi><mo>⋅</mo><mi>a</mi></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p47"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0906" display="block"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>a</mi><mo>−</mo><mn>24</mn><mo>=</mo><mrow><mo>(</mo><mrow><mi>a</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>?</mo></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p48">Consider the factors of 24:</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p49"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0907" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>24</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>⋅</mo><mn>24</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>2</mn><mo>⋅</mo><mn>12</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>3</mn><mo>⋅</mo><mn>8</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#ff0000"><mrow><mn>4</mn><mo>⋅</mo><mn>6</mn></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p50">Suppose we choose the factors 4 and 6 because 4 + 6 = 10, the coefficient of the middle term. Then we have the following incorrect factorization:</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p51"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0908" display="block"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>a</mi><mo>−</mo><mn>24</mn><mover><mo>=</mo><mstyle color="#ff0000"><mo>?</mo></mstyle></mover><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow><mtext> </mtext><mstyle color="#ff0000"><mrow><mi>I</mi><mi>n</mi><mi>c</mi><mi>o</mi><mi>r</mi><mi>r</mi><mi>e</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mi>F</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mrow></mstyle></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p52">When we multiply to check, we find the error.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p53"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0909" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>a</mi><mo>+</mo><mn>4</mn><mi>a</mi><mo>+</mo><mn>24</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>a</mi><mstyle color="#ff0000"><mrow><mtext> </mtext><mo>+</mo><mn>24</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>✗</mi></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p54">In this case, the middle term is correct but the last term is not. Since the last term in the original expression is negative, we need to choose factors that are opposite in sign. Therefore, we must try again. This time we choose the factors −2 and 12 because <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0910" display="inline"><mrow><mo>−</mo><mn>2</mn><mo>+</mo><mn>12</mn><mo>=</mo><mn>10</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p55"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0911" display="block"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>a</mi><mo>−</mo><mn>24</mn><mo>=</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p56">Now the check shows that this factorization is correct.</p>
<p class="para" id="fwk-redden-ch04_s03_s01_p57"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0912" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><mi>a</mi><mo>−</mo><mn>2</mn><mi>a</mi><mo>−</mo><mn>24</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>a</mi><mstyle color="#007f3f"><mrow><mtext> </mtext><mo>−</mo><mtext> </mtext><mn>24</mn></mrow></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s01_p58">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0913" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch04_s03_s01_p59">If we choose the factors wisely, then we can reduce much of the guesswork in this process. However, if a guess is not correct, do not get discouraged; just try a different set of factors. Keep in mind that some polynomials are prime. For example, consider the trinomial <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0914" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>20</mn></mrow></math></span> and the factors of 20:</p>
<p class="para block" id="fwk-redden-ch04_s03_s01_p60"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0915" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>20</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>⋅</mo><mn>20</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>⋅</mo><mn>10</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mo>⋅</mo><mn>5</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s03_s01_p61">There are no factors of 20 whose sum is 3. Therefore, the original trinomial cannot be factored as a product of two binomials with integer coefficients. The trinomial is prime.</p>
</div>
<div class="section" id="fwk-redden-ch04_s03_s02" version="5.0" lang="en">
<h2 class="title editable block">Factoring Trinomials of Higher Degree</h2>
<p class="para editable block" id="fwk-redden-ch04_s03_s02_p01">We can use the trial and error technique to factor trinomials of higher degree.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s02_n01">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch04_s03_s02_p02">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0916" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s02_p03">Begin by factoring the first term <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0917" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s02_p04"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0918" display="block"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mo>=</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s02_p05">Since 5 is prime and the coefficient of the middle term is positive, choose +1 and +5 as the factors of the last term.</p>
<p class="para" id="fwk-redden-ch04_s03_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0919" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s02_p07">Notice that the variable part of the middle term is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0920" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> and the factorization checks out.</p>
<p class="para" id="fwk-redden-ch04_s03_s02_p08"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0921" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s02_p09">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0922" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s03_s02_n02">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch04_s03_s02_p10">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0923" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>4</mn><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>21</mn></mrow></math></span> where <em class="emphasis">n</em> is a positive integer.</p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s02_p11">Begin by factoring the first term <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0924" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><mo>⋅</mo><msup><mi>x</mi><mi>n</mi></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s02_p12"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0925" display="block"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>4</mn><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>21</mn><mo>=</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s02_p13">Factor <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0926" display="inline"><mrow><mo>−</mo><mn>21</mn><mo>=</mo><mn>7</mn><mrow><mo>(</mo><mo>−</mo><mn>3</mn><mo>)</mo></mrow></mrow></math></span> because <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0927" display="inline"><mrow><mn>7</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>=</mo><mo>+</mo><mn>4</mn></mrow></math></span> and write</p>
<p class="para" id="fwk-redden-ch04_s03_s02_p14"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0928" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>4</mn><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>21</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s02_p15">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0929" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span> The check is left to the reader.</p>
</div>
<div class="callout block" id="fwk-redden-ch04_s03_s02_n02a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s03_s02_p16"><strong class="emphasis bold">Try this!</strong> Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0930" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>42</mn></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s02_p17">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0931" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
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</div>
</div>
<div class="section" id="fwk-redden-ch04_s03_s03" version="5.0" lang="en">
<h2 class="title block">Factoring Trinomials of the Form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0932" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span>
</h2>
<p class="para block" id="fwk-redden-ch04_s03_s03_p01">Factoring trinomials of the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0933" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> can be challenging because the middle term is affected by the factors of both <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0934" display="inline"><mi>a</mi></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0935" display="inline"><mi>c</mi><mo>.</mo></math></span> In general,</p>
<p class="para block" id="fwk-redden-ch04_s03_s03_p02"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0936" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mi>a</mi></mstyle><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mstyle color="#007fbf"><mi>b</mi></mstyle><mi>x</mi><mo>+</mo><mstyle color="#007fbf"><mi>c</mi></mstyle></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>p</mi><mi>x</mi><mo>+</mo><mi>m</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>q</mi><mi>x</mi><mo>+</mo><mi>n</mi></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mi>p</mi><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>p</mi><mi>n</mi><mi>x</mi><mo>+</mo><mi>q</mi><mi>m</mi><mi>x</mi><mo>+</mo><mi>m</mi><mi>n</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mrow><mi>p</mi><mi>q</mi></mrow></mstyle><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mstyle color="#007fbf"><mrow><mo stretchy="false">(</mo><mi>p</mi><mi>n</mi><mo>+</mo><mi>q</mi><mi>m</mi><mo stretchy="false">)</mo></mrow></mstyle><mi>x</mi><mo>+</mo><mstyle color="#007fbf"><mrow><mi>m</mi><mi>n</mi></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s03_s03_p03">This gives us,</p>
<p class="para block" id="fwk-redden-ch04_s03_s03_p04"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0937" display="block"><mrow><mi>a</mi><mo>=</mo><mi>p</mi><mi>q</mi><mtext> and</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>b</mi><mo>=</mo><mi>p</mi><mi>n</mi><mo>+</mo><mi>q</mi><mi>m</mi><mo>,</mo><mtext> where</mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>c</mi><mo>=</mo><mi>m</mi><mi>n</mi></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s03_s03_p05">In short, when the leading coefficient of a trinomial is something other than 1, there will be more to consider when determining the factors using the trial and error method. The key lies in the understanding of how the middle term is obtained. Multiply <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0938" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</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></math></span> and carefully follow the formation of the middle term.</p>
<div class="informalfigure large block">
<img src="section_07/d9fa1fbde0e4ce9c818afce958ed4fd6.png">
</div>
<p class="para editable block" id="fwk-redden-ch04_s03_s03_p07">As we have seen before, the product of the first terms of each binomial is equal to the first term of the trinomial. The middle term of the trinomial is the sum of the products of the outer and inner terms of the binomials. The product of the last terms of each binomial is equal to the last term of the trinomial. Visually, we have the following:</p>
<div class="informalfigure large block">
<img src="section_07/df95fd38b1436d2535ca0588e40aa3e3.png">
</div>
<p class="para block" id="fwk-redden-ch04_s03_s03_p09">For this reason, we need to look for products of the factors of the first and last terms whose sum is equal to the coefficient of the middle term. For example, to factor <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0939" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>29</mn><mi>x</mi><mo>+</mo><mn>35</mn></mrow></math></span>, look at the factors of 6 and 35.</p>
<p class="para block" id="fwk-redden-ch04_s03_s03_p10"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0940" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mn>6</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>⋅</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow><mn>35</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>⋅</mo><mn>35</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>2</mn><mo>⋅</mo><mn>3</mn></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>5</mn><mo>⋅</mo><mn>7</mn></mrow></mstyle></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch04_s03_s03_p11">The combination that produces the coefficient of the middle term is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0941" display="inline"><mrow><mn>2</mn><mo>⋅</mo><mn>7</mn><mo>+</mo><mn>3</mn><mo>⋅</mo><mn>5</mn><mo>=</mo><mn>14</mn><mo>+</mo><mn>15</mn><mo>=</mo><mn>29</mn></mrow><mo>.</mo></math></span> Make sure that the outer terms have coefficients 2 and 7, and that the inner terms have coefficients 5 and 3. Use this information to factor the trinomial.</p>
<p class="para block" id="fwk-redden-ch04_s03_s03_p12"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0942" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>29</mn><mi>x</mi><mo>+</mo><mn>35</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><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><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mtext> </mtext></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch04_s03_s03_p13">We can always check by multiplying; this is left to the reader.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s03_n01">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch04_s03_s03_p14">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0943" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p15">Since the leading coefficient and the last term are both prime, there is only one way to factor each.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p16"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0944" display="block"><mrow><mn>5</mn><mo>=</mo><mn>1</mn><mo>⋅</mo><mn>5</mn><mtext> and </mtext><mn>3</mn><mo>=</mo><mn>1</mn><mo>⋅</mo><mn>3</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p17">Begin by writing the factors of the first term, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0945" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span>, as follows:</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p18"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0946" display="block"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mo stretchy="false">(</mo><mn>5</mn><mi>x</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle><mo stretchy="false">)</mo></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p19">The middle and last term are both positive; therefore, the factors of 3 are chosen as positive numbers. In this case, the only choice is in which grouping to place these factors.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p20"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0947" display="block"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mrow><mo>)</mo></mrow><mtext> or </mtext><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p21">Determine which grouping is correct by multiplying each expression.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p22"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0948" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>5</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#ff0000"><mo>✗</mo></mstyle></mrow></mtd></mtr></mtable></mtd></mtr><mtr><mtd columnalign="right"><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mi>y</mi><mo>+</mo><mn>15</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr></mtable></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p23">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0949" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s03_s03_n02">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch04_s03_s03_p24">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0950" display="inline"><mrow><mn>18</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>−</mo><mn>4</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p25">First, consider the factors of the coefficients of the first and last terms.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p26"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0951" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>18</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo>⋅</mo><mn>18</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mn>4</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>1</mn><mo>⋅</mo><mn>4</mn></mrow></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>2</mn><mo>⋅</mo><mn>9</mn></mrow></mstyle></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>⋅</mo><mn>2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>3</mn><mo>⋅</mo><mn>6</mn></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p27">We are searching for products of factors whose sum equals the coefficient of the middle term, −1. After some thought, we can see that the sum of 8 and −9 is −1 and the combination that gives this follows:</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p28"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0952" display="block"><mrow><mn>2</mn><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mo>+</mo><mn>9</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>8</mn><mo>−</mo><mn>9</mn><mo>=</mo><mo>−</mo><mn>1</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p29">Factoring begins at this point with two sets of blank parentheses.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p30"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0953" display="block"><mrow><mn>18</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>−</mo><mn>4</mn><mo>=</mo><mrow><mo>(</mo><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p31">Use <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0954" display="inline"><mrow><mn>2</mn><mi>a</mi><mi>b</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0955" display="inline"><mrow><mn>9</mn><mi>a</mi><mi>b</mi></mrow></math></span> as factors of <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0956" display="inline"><mrow><mn>18</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p32"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0957" display="block"><mrow><mn>18</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>−</mo><mn>4</mn><mo>=</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mi>b</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>9</mn><mi>a</mi><mi>b</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p33">Next use the factors 1 and 4 in the correct order so that the inner and outer products are <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0958" display="inline"><mrow><mo>−</mo><mn>9</mn><mi>a</mi><mi>b</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0959" display="inline"><mrow><mn>8</mn><mi>a</mi><mi>b</mi></mrow></math></span> respectively.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p34"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0960" display="block"><mrow><mn>18</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>−</mo><mn>4</mn><mo>=</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mi>b</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>9</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p35">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0961" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>a</mi><mi>b</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>9</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> The complete check is left to the reader.</p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s03_s03_p36">It is a good practice to first factor out the GCF, if there is one. Doing this produces a trinomial factor with smaller coefficients. As we have seen, trinomials with smaller coefficients require much less effort to factor. This commonly overlooked step is worth identifying early.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s03_n03">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch04_s03_s03_p37">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0962" display="inline"><mrow><mn>12</mn><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>26</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>y</mi></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p38">Begin by factoring out the GCF.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p39"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0963" display="block"><mrow><mn>12</mn><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>26</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>y</mi><mo>=</mo><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>13</mn><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p40">After factoring out <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0964" display="inline"><mrow><mn>2</mn><mi>y</mi></mrow></math></span>, the coefficients of the resulting trinomial are smaller and have fewer factors. We can factor the resulting trinomial using <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0965" display="inline"><mrow><mn>6</mn><mo>=</mo><mn>2</mn><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0966" display="inline"><mrow><mn>5</mn><mo>=</mo><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Notice that these factors can produce −13 in two ways:</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p41"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0967" display="block"><mrow><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mn>2</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>10</mn><mo>−</mo><mn>3</mn><mo>=</mo><mo>−</mo><mn>13</mn></mrow></mtd></mtr><mtr><mtd columnalign="right"><mrow><mn>2</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mn>1</mn></mstyle><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mrow><mo>(</mo><mstyle color="#007f3f"><mrow><mo>−</mo><mn>5</mn></mrow></mstyle><mo>)</mo></mrow></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo>−</mo><mn>15</mn><mo>=</mo><mo>−</mo><mn>13</mn></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p42">Because the last term is −5, the correct combination requires the factors 1 and 5 to be opposite signs. Here we use 2(1) = 2 and 3(−5) = −15 because the sum is −13 and the product of (1)(−5) = −5.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p43"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0968" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>12</mn><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>26</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>y</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>13</mn><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>?</mo></mstyle></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p44">Check.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p45"><span class="informalequation">
<math xml:id="fwk-redden-ch04_m0969" display="block"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="right"><mrow><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>y</mi><mo>−</mo><mn>15</mn><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>13</mn><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="right"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>12</mn><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>26</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>y</mi><mi> </mi><mi> </mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mrow></mtd></mtr></mtable></mrow></math>
</span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p46">The factor <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0970" display="inline"><mrow><mn>2</mn><mi>y</mi></mrow></math></span> is part of the factored form of the original expression; be sure to include it in the answer.</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p47">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0971" display="inline"><mrow><mn>2</mn><mi>y</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s03_s03_p48">It is a good practice to consistently work with trinomials where the leading coefficient is positive. If the leading coefficient is negative, factor it out along with any GCF. Note that sometimes the factor will be −1.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s03_n04">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch04_s03_s03_p49">Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0972" display="inline"><mrow><mo>−</mo><mn>18</mn><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><mn>69</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p50">In this example, the GCF is <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0973" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span> Because the leading coefficient is negative we begin by factoring out <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0974" display="inline"><mrow><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p51"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0975" display="block"><mrow><mo>−</mo><mn>18</mn><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><mn>69</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>23</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p52">At this point, factor the remaining trinomial as usual, remembering to write the <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0976" display="inline"><mrow><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> as a factor in the final answer. Use 6 = 1(6) and −4 = 4(−1) because <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0977" display="inline"><mrow><mn>1</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mn>6</mn><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mo>=</mo><mn>23</mn></mrow><mo>.</mo></math></span> Therefore,</p>
<p class="para" id="fwk-redden-ch04_s03_s03_p53"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0978" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mo>−</mo><mn>18</mn><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><mn>69</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>23</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p54">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0979" display="inline"><mrow><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> The check is left to the reader.</p>
</div>
<div class="callout block" id="fwk-redden-ch04_s03_s03_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch04_s03_s03_p55"><strong class="emphasis bold">Try this!</strong> Factor: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0980" display="inline"><mrow><mo>−</mo><mn>12</mn><msup><mi>a</mi><mn>5</mn></msup><mi>b</mi><mo>+</mo><msup><mi>a</mi><mn>3</mn></msup><msup><mi>b</mi><mn>3</mn></msup><mo>+</mo><mi>a</mi><msup><mi>b</mi><mn>5</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s03_p56">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0981" display="inline"><mrow><mo>−</mo><mi>a</mi><mi>b</mi><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
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<div class="section" id="fwk-redden-ch04_s03_s04" version="5.0" lang="en">
<h2 class="title editable block">Factoring Using the AC Method</h2>
<p class="para block" id="fwk-redden-ch04_s03_s04_p01">An alternate technique for factoring trinomials, called the <span class="margin_term"><a class="glossterm">AC method</a><span class="glossdef">Method used for factoring trinomials by replacing the middle term with two terms that allow us to factor the resulting four-term polynomial by grouping.</span></span>, makes use of the grouping method for factoring four-term polynomials. If a trinomial in the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0982" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> can be factored, then the middle term, <em class="emphasis">bx</em>, can be replaced with two terms with coefficients whose sum is <em class="emphasis">b</em> and product is <em class="emphasis">ac</em>. This substitution results in an equivalent expression with four terms that can be factored by grouping.</p>
<div class="callout block" id="fwk-redden-ch04_s03_s04_n01">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch04_s03_s04_p02">Factor using the AC method: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0983" display="inline"><mrow><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>31</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s04_p03">Here <em class="emphasis">a</em> = 18, <em class="emphasis">b</em> = −31, and <em class="emphasis">c</em> = 6.</p>
<p class="para" id="fwk-redden-ch04_s03_s04_p04"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0984" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>a</mi><mi>c</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>18</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>108</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s04_p05">Factor 108, and search for factors whose sum is −31.</p>
<p class="para" id="fwk-redden-ch04_s03_s04_p06"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0985" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>108</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>1</mn><mo stretchy="false">(</mo><mo>−</mo><mn>108</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>2</mn><mo stretchy="false">(</mo><mo>−</mo><mn>54</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>3</mn><mo stretchy="false">(</mo><mo>−</mo><mn>36</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mo>−</mo><mn>4</mn><mo stretchy="false">(</mo><mo>−</mo><mn>27</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>6</mn><mo stretchy="false">(</mo><mo>−</mo><mn>18</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>−</mo><mn>9</mn><mo stretchy="false">(</mo><mo>−</mo><mn>12</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s04_p07">In this case, the sum of the factors −27 and −4 equals the middle coefficient, −31. Therefore, <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0986" display="inline"><mrow><mo>−</mo><mn>31</mn><mi>x</mi><mo>=</mo><mo>−</mo><mn>27</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>x</mi></mrow></math></span>, and we can write</p>
<p class="para" id="fwk-redden-ch04_s03_s04_p08"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0987" display="block"><mrow><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mstyle color="#007f3f"><mo>−</mo><mn>31</mn><mi>x</mi></mstyle><mo>+</mo><mn>6</mn><mo>=</mo><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mstyle color="#007f3f"><mo>−</mo><mn>27</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>x</mi></mstyle><mo>+</mo><mn>6</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s04_p09">Factor the equivalent expression by grouping.</p>
<p class="para" id="fwk-redden-ch04_s03_s04_p10"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0988" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>31</mn><mi>x</mi><mo>+</mo><mn>6</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>27</mn><mi>x</mi><mo>−</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>6</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn><mi>x</mi><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>9</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s04_p11">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0989" display="inline"><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>9</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch04_s03_s04_n02">
<h3 class="title">Example 12</h3>
<p class="para" id="fwk-redden-ch04_s03_s04_p12">Factor using the AC method: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0990" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>15</mn></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch04_s03_s04_p13">Here <em class="emphasis">a</em> = 4, <em class="emphasis">b</em> = −7, and <em class="emphasis">c</em> = −15.</p>
<p class="para" id="fwk-redden-ch04_s03_s04_p14"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0991" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mi>a</mi><mi>c</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><mo stretchy="false">(</mo><mo>−</mo><mn>15</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>60</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s04_p15">Factor −60 and search for factors whose sum is −7.</p>
<p class="para" id="fwk-redden-ch04_s03_s04_p16"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0992" display="block"><mrow><mtable columnspacing="0.1em" columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>−</mo><mn>60</mn></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>1</mn><mo stretchy="false">(</mo><mo>−</mo><mn>60</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>2</mn><mo stretchy="false">(</mo><mo>−</mo><mn>30</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>3</mn><mo stretchy="false">(</mo><mo>−</mo><mn>20</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>4</mn><mo stretchy="false">(</mo><mo>−</mo><mn>15</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007f3f"><mrow><mn>5</mn><mo stretchy="false">(</mo><mo>−</mo><mn>12</mn><mo stretchy="false">)</mo></mrow></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mo>✓</mo></mstyle></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mn>6</mn><mrow><mo>(</mo><mrow><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mrow></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s04_p17">The sum of factors 5 and −12 equals the middle coefficient, −7. Replace <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0993" display="inline"><mrow><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi></mrow></math></span> with <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0994" display="inline"><mrow><mn>5</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>12</mn><mi>x</mi><mi>y</mi></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s04_p18"><span class="informalequation"><math xml:id="fwk-redden-ch04_m0995" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>15</mn></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>12</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>15</mn><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>F</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mi>g</mi><mi>r</mi><mi>o</mi><mi>u</mi><mi>p</mi><mi>i</mi><mi>n</mi><mi>g</mi><mtext>.</mtext></mstyle><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>x</mi><mi>y</mi><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><mn>3</mn><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch04_s03_s04_p19">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0996" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> The check is left to the reader.</p>
</div>
<p class="para editable block" id="fwk-redden-ch04_s03_s04_p20">If factors of <em class="emphasis">ac</em> cannot be found to add up to <em class="emphasis">b</em> then the trinomial is prime.</p>
<div class="key_takeaways block" id="fwk-redden-ch04_s03_s04_n03">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch04_s03_s04_l01" mark="bullet">
<li>If a trinomial of the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0997" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> factors into the product of two binomials, then the coefficient of the middle term is the sum of factors of the last term.</li>
<li>If a trinomial of the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0998" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> factors into the product of two binomials, then the coefficient of the middle term will be the sum of certain products of factors of the first and last terms.</li>
<li>If the trinomial has a greatest common factor, then it is a best practice to first factor out the GCF before attempting to factor it into a product of binomials.</li>
<li>If the leading coefficient of a trinomial is negative, then it is a best practice to first factor that negative factor out before attempting to factor the trinomial.</li>
<li>Factoring is one of the more important skills required in algebra. For this reason, you should practice working as many problems as it takes to become proficient.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch04_s03_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s03_qs01_qd01">
<h3 class="title">Part A: Factoring Trinomials of the Form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m0999" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span>
</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s03_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch04_s03_qs01_p01"><strong class="emphasis bold">Factor.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1000" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1002" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1004" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1006" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1008" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>14</mn><mi>x</mi><mo>+</mo><mn>48</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1010" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>15</mn><mi>x</mi><mo>+</mo><mn>54</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p14"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1012" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>11</mn><mi>x</mi><mo>−</mo><mn>30</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p16"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1013" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p18"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1014" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn><mi>x</mi><mo>+</mo><mn>81</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p20"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1016" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>22</mn><mi>x</mi><mo>+</mo><mn>121</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p22"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1018" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mi>y</mi><mo>−</mo><mn>20</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p24"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1020" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p26"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1022" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>50</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1024" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>16</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>48</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1026" display="inline"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>a</mi><mi>b</mi><mo>−</mo><mn>72</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1028" display="inline"><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>21</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>80</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1030" display="inline"><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mn>14</mn><mi>u</mi><mi>v</mi><mo>−</mo><mn>32</mn><msup><mi>v</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1032" display="inline"><mrow><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>m</mi><mi>n</mi><mo>−</mo><mn>98</mn><msup><mi>n</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1034" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1036" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>2</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1038" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p44"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1040" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>13</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>30</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p46"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1042" display="inline"><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>48</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p48"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1044" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>25</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p50"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1046" display="inline"><mrow><msup><mi>y</mi><mn>4</mn></msup><mo>−</mo><mn>20</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>100</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p52"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1048" display="inline"><mrow><msup><mi>y</mi><mn>4</mn></msup><mo>+</mo><mn>14</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>49</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p54"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1050" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p56"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1052" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>15</mn><msup><mi>y</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p58"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1054" display="inline"><mrow><msup><mi>a</mi><mn>4</mn></msup><msup><mi>b</mi><mn>4</mn></msup><mo>−</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p60"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1056" display="inline"><mrow><msup><mi>a</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>b</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p62"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1058" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><mn>18</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>40</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p64"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1060" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mn>18</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>45</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p66"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1062" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>6</mn><msup><mi>y</mi><mn>6</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p68"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1064" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>20</mn><msup><mi>y</mi><mn>6</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p70"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1066" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><msup><mi>y</mi><mn>6</mn></msup><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p72"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1068" display="inline"><mrow><msup><mi>x</mi><mn>6</mn></msup><msup><mi>y</mi><mn>6</mn></msup><mo>+</mo><mn>16</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>48</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p74"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1070" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>12</mn><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><mn>32</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p76"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1072" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>41</mn><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><mn>40</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p78"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1074" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>2</mn><mi>a</mi><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p80"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1076" display="inline"><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><mn>2</mn><mi>a</mi><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s03_qs01_qd02">
<h3 class="title">Part B: Factoring Trinomials of the Form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1078" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span>
</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s03_qs01_qd02_qd01" start="41">
<p class="para" id="fwk-redden-ch04_s03_qs01_p82"><strong class="emphasis bold">Factor.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p83"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1079" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p85"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1081" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>9</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p87"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1083" display="inline"><mrow><mn>6</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>13</mn><mi>a</mi><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p89"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1085" display="inline"><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>11</mn><mi>a</mi><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p91"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1087" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>−</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p93"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1089" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p95"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1091" display="inline"><mrow><mn>24</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>35</mn><mi>y</mi><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p97"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1093" display="inline"><mrow><mn>10</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>23</mn><mi>y</mi><mo>+</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p99"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1095" display="inline"><mrow><mn>14</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>11</mn><mi>x</mi><mo>+</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p101"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1096" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p103"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1097" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>28</mn><mi>x</mi><mo>+</mo><mn>49</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p105"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1099" display="inline"><mrow><mn>36</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>60</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p107"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1101" display="inline"><mrow><mn>27</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa54">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p109"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1103" display="inline"><mrow><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>17</mn><mi>x</mi><mo>−</mo><mn>20</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p111"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1105" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>23</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p113"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1107" display="inline"><mrow><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>21</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>27</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p115"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1109" display="inline"><mrow><mn>8</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>9</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p117"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1111" display="inline"><mrow><mn>12</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>−</mo><mn>20</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p119"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1113" display="inline"><mrow><mn>8</mn><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mn>26</mn><mi>u</mi><mi>v</mi><mo>+</mo><mn>15</mn><msup><mi>v</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p121"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1115" display="inline"><mrow><mn>24</mn><msup><mi>m</mi><mn>2</mn></msup><mo>−</mo><mn>26</mn><mi>m</mi><mi>n</mi><mo>+</mo><mn>5</mn><msup><mi>n</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p123"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1117" display="inline"><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>9</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p125"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1119" display="inline"><mrow><mn>16</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>40</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>25</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p127"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1121" display="inline"><mrow><mn>5</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>9</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p129"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1123" display="inline"><mrow><mn>7</mn><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>15</mn><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>−</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p131"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1125" display="inline"><mrow><mn>7</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>22</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p133"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1127" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>41</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p135"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1129" display="inline"><mrow><mn>4</mn><msup><mi>y</mi><mn>6</mn></msup><mo>−</mo><mn>3</mn><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p137"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1131" display="inline"><mrow><mn>12</mn><msup><mi>y</mi><mn>6</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p139"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1133" display="inline"><mrow><mn>5</mn><msup><mi>a</mi><mn>4</mn></msup><msup><mi>b</mi><mn>4</mn></msup><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>18</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p141"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1135" display="inline"><mrow><mn>21</mn><msup><mi>a</mi><mn>4</mn></msup><msup><mi>b</mi><mn>4</mn></msup><mo>+</mo><mn>5</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p143"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1137" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>6</mn></msup><msup><mi>y</mi><mn>6</mn></msup><mo>+</mo><mn>17</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>10</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p145"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1139" display="inline"><mrow><mn>16</mn><msup><mi>x</mi><mn>6</mn></msup><msup><mi>y</mi><mn>6</mn></msup><mo>+</mo><mn>46</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>15</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p147"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1141" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><mn>10</mn><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>25</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p149"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1143" display="inline"><mrow><mn>30</mn><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><mn>11</mn><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p151"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1145" display="inline"><mrow><mn>36</mn><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>12</mn><mi>a</mi><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p153"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1147" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><mn>12</mn><mi>a</mi><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p155"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1149" display="inline"><mrow><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>14</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p157"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1151" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>13</mn><mi>x</mi><mo>−</mo><mn>20</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa79">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p159"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1153" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>24</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa80">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p161"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1155" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>48</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa81">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p163"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1157" display="inline"><mrow><mn>54</mn><mo>−</mo><mn>12</mn><mi>x</mi><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa82">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p165"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1159" display="inline"><mrow><mn>60</mn><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p167"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1161" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>16</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>20</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p169"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1163" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>12</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>14</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p171"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1165" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>−</mo><mn>24</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p173"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1167" display="inline"><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><mi>y</mi><mo>−</mo><mn>6</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p175"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1169" display="inline"><mrow><mn>4</mn><msup><mi>a</mi><mn>3</mn></msup><mi>b</mi><mo>−</mo><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>24</mn><mi>a</mi><msup><mi>b</mi><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p177"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1171" display="inline"><mrow><mn>15</mn><msup><mi>a</mi><mn>4</mn></msup><mi>b</mi><mo>−</mo><mn>33</mn><msup><mi>a</mi><mn>3</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p179"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1173" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>5</mn></msup><mi>y</mi><mo>+</mo><mn>30</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>75</mn><mi>x</mi><msup><mi>y</mi><mn>5</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p181"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1175" display="inline"><mrow><mn>45</mn><msup><mi>x</mi><mn>5</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>60</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>4</mn></msup><mo>+</mo><mn>20</mn><mi>x</mi><msup><mi>y</mi><mn>6</mn></msup></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s03_qs01_qd02_qd02" start="91">
<p class="para" id="fwk-redden-ch04_s03_qs01_p183"><strong class="emphasis bold">Factor.</strong></p>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa91">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1177" display="inline"><mrow><mn>4</mn><mo>−</mo><mn>25</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa92">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p186"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1179" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><msup><mi>y</mi><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa93">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p188"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1181" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa94">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p190"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1183" display="inline"><mrow><mn>30</mn><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>57</mn><mi>a</mi><mi>b</mi><mo>−</mo><mn>6</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa95">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p192"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1185" display="inline"><mrow><mn>10</mn><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>a</mi><mo>−</mo><mn>6</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>3</mn><mi>b</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa96">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p194"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1187" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>x</mi><mo>−</mo><mn>12</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa97">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p196"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1189" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa98">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p198"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1190" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa99">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p200"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1191" display="inline"><mrow><mn>15</mn><msup><mi>a</mi><mn>3</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><mi>a</mi><msup><mi>b</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa100">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p202"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1193" display="inline"><mrow><mn>54</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>63</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch04_s03_qs01_qd03">
<h3 class="title">Part D: Discussion Board</h3>
<ol class="qandadiv" id="fwk-redden-ch04_s03_qs01_qd03_qd01" start="101">
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa101">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p204">Create your own trinomial of the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1195" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> that factors. Share it, along with the solution, on the discussion board.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa102">
<div class="question">
<p class="para" id="fwk-redden-ch04_s03_qs01_p205">Create a trinomial of the form <span class="inlineequation"><math xml:id="fwk-redden-ch04_m1196" display="inline"><mrow><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></math></span> that does not factor and share it along with the reason why it does not factor.</p>
</div>
</li>
</ol>
</ol>
</div>
<div class="qandaset block" id="fwk-redden-ch04_s03_qs01_ans" defaultlabel="number">
<h3 class="title">Answers</h3>
<ol class="qandadiv">
<li class="qandaentry" id="fwk-redden-ch04_s03_qs01_qa01_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p03_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1001" display="inline"><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>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa03_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p07_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1005" display="inline"><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>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa05_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p11_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1009" display="inline"><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>8</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa07_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p15_ans">Prime</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa09_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p19_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1015" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa11_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p23_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1019" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn><mi>y</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa13_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p27_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1023" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mi>y</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa15_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p31_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1027" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mn>6</mn><mi>b</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>12</mn><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa17_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p35_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1031" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>−</mo><mn>2</mn><mi>v</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>u</mi><mo>+</mo><mn>16</mn><mi>v</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa19_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p39_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1035" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa21_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p43_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1039" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa23_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p47_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1043" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa25_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p51_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1047" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa27_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p55_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1051" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa29_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p59_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1055" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa31_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p63_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1059" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>20</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa33_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p67_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1063" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>y</mi><mn>3</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa35_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p71_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1067" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa37_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p75_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1071" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch04_s03_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_s03_qs01_qa39_ans">
<div class="answer">
<p class="para" id="fwk-redden-ch04_s03_qs01_p79_ans"><span class="inlineequation"><math xml:id="fwk-redden-ch04_m1075" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><mi>a</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>