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<div class="section" id="fwk-redden-ch01_s06" version="5.0" lang="en">
<h2 class="title editable block">
<span class="title-prefix">1.6</span> Polynomials and Their Operations</h2>
<div class="learning_objectives editable block" id="fwk-redden-ch01_s06_n01">
<h3 class="title">Learning Objectives</h3>
<ol class="orderedlist" id="fwk-redden-ch01_s06_o01" numeration="arabic">
<li>Identify a polynomial and determine its degree.</li>
<li>Add and subtract polynomials.</li>
<li>Multiply and divide polynomials.</li>
</ol>
</div>
<div class="section" id="fwk-redden-ch01_s06_s01" version="5.0" lang="en">
<h2 class="title editable block">Definitions</h2>
<p class="para editable block" id="fwk-redden-ch01_s06_s01_p01">A <span class="margin_term"><a class="glossterm">polynomial</a><span class="glossdef">An algebraic expression consisting of terms with real number coefficients and variables with whole number exponents.</span></span> is a special algebraic expression with terms that consist of real number coefficients and variable factors with whole number exponents. Some examples of polynomials follow:</p>
<p class="para block" id="fwk-redden-ch01_s06_s01_p02">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1346" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1347" display="inline"><mrow><mn>7</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>5</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1348" display="inline"><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1349" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>−</mo><mn>4</mn><mi>x</mi><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>7</mn></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch01_s06_s01_p03">The <span class="margin_term"><a class="glossterm">degree of a term</a><span class="glossdef">The exponent of the variable. If there is more than one variable in the term, the degree of the term is the sum their exponents.</span></span> in a polynomial is defined to be the exponent of the variable, or if there is more than one variable in the term, the degree is the sum of their exponents. Recall that <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1350" display="inline"><mrow><msup><mi>x</mi><mn>0</mn></msup><mo>=</mo><mn>1</mn></mrow></math></span>; any constant term can be written as a product of <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1351" display="inline"><mrow><msup><mi>x</mi><mn>0</mn></msup></mrow></math></span> and itself. Hence the degree of a constant term is 0.</p>
<p class="para block" id="fwk-redden-ch01_s06_s01_p04">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Term</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Degree</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1352" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1353" display="inline"><mn>2</mn></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1354" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1355" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>=</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1356" display="inline"><mrow><mn>7</mn><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>3</mn></msup></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1357" display="inline"><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mo>=</mo><mn>5</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1358" display="inline"><mn>8</mn></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1359" display="inline"><mn>0</mn></math></span>, since <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1360" display="inline"><mrow><mn>8</mn><mo>=</mo><mn>8</mn><msup><mi>x</mi><mn>0</mn></msup></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1361" display="inline"><mrow><mn>2</mn><mi>x</mi></mrow></math></span></p></td>
<td align="center"><p class="para">1, since <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1362" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>1</mn></msup></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s01_p05">The <span class="margin_term"><a class="glossterm">degree of a polynomial</a><span class="glossdef">The largest degree of all of its terms.</span></span> is the largest degree of all of its terms.</p>
<p class="para block" id="fwk-redden-ch01_s06_s01_p06">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Polynomial</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Degree</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1363" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>5</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1364" display="inline"><mn>5</mn></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1365" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>−</mo><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>7</mn></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1366" display="inline"><mn>4</mn></math></span>, because <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1367" display="inline"><mrow><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>3</mn></msup></mrow></math></span> has degree 4.</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1368" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>5</mn><mn>4</mn></mfrac></mrow></math></span></p></td>
<td align="center"><p class="para">1, because <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1369" display="inline"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>1</mn></msup></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch01_s06_s01_p07">Of particular interest are <span class="margin_term"><a class="glossterm">polynomials with one variable</a><span class="glossdef">A polynomial where each term has the form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1370" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><msup><mi>x</mi><mi>n</mi></msup></mrow></math></span>, where <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1371" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></math></span> is any real number and <em class="emphasis">n</em> is any whole number.</span></span>, where each term is of the form <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1372" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub><msup><mi>x</mi><mi>n</mi></msup></mrow><mo>.</mo></math></span> Here <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1373" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></math></span> is any real number and <em class="emphasis">n</em> is any whole number. Such polynomials have the standard form:</p>
<p class="para block" id="fwk-redden-ch01_s06_s01_p08"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1374" display="block"><mrow><msub><mi>a</mi><mi>n</mi></msub><msup><mi>x</mi><mi>n</mi></msup><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msup><mi>x</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s06_s01_p09">Typically, we arrange terms of polynomials in descending order based on the degree of each term. The <span class="margin_term"><a class="glossterm">leading coefficient</a><span class="glossdef">The coefficient of the term with the largest degree.</span></span> is the coefficient of the variable with the highest power, in this case, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1375" display="inline"><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow><mo>.</mo></math></span></p>
<div class="callout block" id="fwk-redden-ch01_s06_s01_n01">
<h3 class="title">Example 1</h3>
<p class="para" id="fwk-redden-ch01_s06_s01_p10">Write in standard form: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1376" display="inline"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>7</mn><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s01_p11">Since terms are defined to be separated by addition, we write the following:</p>
<p class="para" id="fwk-redden-ch01_s06_s01_p12"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1377" display="block"><mtable columnalign="left"><mtr><mtd columnalign="left"><mn>3</mn><mi>x</mi><mo>−</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>7</mn><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>4</mn></msup></mtd></mtr><mtr><mtd columnalign="left"><mo>=</mo><mn>3</mn><mi>x</mi><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>7</mn><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><msup><mi>x</mi><mn>4</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s01_p13">In this form, we can see that the subtraction in the original corresponds to negative coefficients. Because addition is commutative, we can write the terms in descending order based on the degree as follows:</p>
<p class="para" id="fwk-redden-ch01_s06_s01_p14"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1378" display="block"><mtable columnalign="left"><mtr><mtd><mo>=</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>4</mn></mrow><mo>)</mo></mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mtd></mtr><mtr><mtd><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>5</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>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s01_p15">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1379" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>5</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>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s01_p16">We classify polynomials by the number of terms and the degree:</p>
<p class="para block" id="fwk-redden-ch01_s06_s01_p17">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Expression</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Classification</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Degree</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1380" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>7</mn></msup></mrow></math></span></p></td>
<td align="left"><p class="para"><strong class="emphasis bold">Monomial</strong> (one term)</p></td>
<td align="center"><p class="para">7</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1381" display="inline"><mrow><mn>8</mn><msup><mi>x</mi><mn>6</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p></td>
<td align="left"><p class="para"><strong class="emphasis bold">Binomial</strong> (two terms)</p></td>
<td align="center"><p class="para">6</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1382" display="inline"><mrow><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>1</mn><mi> </mi></mrow></math></span></p></td>
<td align="left"><p class="para"><strong class="emphasis bold">Trinomial</strong> (three terms)</p></td>
<td align="center"><p class="para">2</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1383" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow></math></span></p></td>
<td align="left"><p class="para"><strong class="emphasis bold">Polynomial</strong> (many terms)</p></td>
<td align="center"><p class="para">3</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">Polynomial with one term.</span></span></p>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">Polynomial with two terms.</span></span></p>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">Polynomial with three terms.</span></span></p>
<p class="para editable block" id="fwk-redden-ch01_s06_s01_p18">We can further classify polynomials with one variable by their degree:</p>
<p class="para block" id="fwk-redden-ch01_s06_s01_p19">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<thead>
<tr>
<th align="center"><p class="para"><em class="emphasis">Polynomial</em></p></th>
<th align="center"><p class="para"><em class="emphasis">Name</em></p></th>
</tr>
</thead>
<tbody>
<tr>
<td align="center"><p class="para">5</p></td>
<td align="left"><p class="para"><strong class="emphasis bold">Constant</strong> (degree 0)</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1384" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p></td>
<td align="left"><p class="para"><strong class="emphasis bold">Linear</strong> (degree 1)</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1385" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p></td>
<td align="left"><p class="para"><strong class="emphasis bold">Quadratic</strong> (degree 2)</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1386" display="inline"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow></math></span></p></td>
<td align="left"><p class="para"><strong class="emphasis bold">Cubic</strong> (degree 3)</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1387" display="inline"><mrow><mn>7</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow></math></span></p></td>
<td align="left"><p class="para">Fourth-degree polynomial</p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">A polynomial with degree 0.</span></span></p>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">A polynomial with degree 1.</span></span></p>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">A polynomial with degree 2.</span></span></p>
<p class="para block"><span class="margin_term"><a class="glossterm"></a><span class="glossdef">A polynomial with degree 3.</span></span></p>
<p class="para block" id="fwk-redden-ch01_s06_s01_p20">In this text, we call any polynomial of degree <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1388" display="inline"><mrow><mi>n</mi><mo>≥</mo><mn>4</mn></mrow></math></span> an <em class="emphasis">n</em>th-degree polynomial. In other words, if the degree is 4, we call the polynomial a fourth-degree polynomial. If the degree is 5, we call it a fifth-degree polynomial, and so on.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s01_n02">
<h3 class="title">Example 2</h3>
<p class="para" id="fwk-redden-ch01_s06_s01_p21">State whether the following polynomial is linear or quadratic and give the leading coefficient: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1389" display="inline"><mrow><mn>25</mn><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><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-ch01_s06_s01_p22">The highest power is 2; therefore, it is a quadratic polynomial. Rewriting in standard form we have</p>
<p class="para" id="fwk-redden-ch01_s06_s01_p23"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1390" display="block"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s01_p24">Here <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1391" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mo>−</mo><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> and thus the leading coefficient is −1.</p>
<p class="para" id="fwk-redden-ch01_s06_s01_p25">Answer: Quadratic; leading coefficient: −1</p>
</div>
</div>
<div class="section" id="fwk-redden-ch01_s06_s02" version="5.0" lang="en">
<h2 class="title editable block">Adding and Subtracting Polynomials</h2>
<p class="para block" id="fwk-redden-ch01_s06_s02_p01">We begin by simplifying algebraic expressions that look like <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1392" display="inline"><mrow><mo>+</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span> or <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1393" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span> Here, the coefficients are actually implied to be +1 and −1 respectively and therefore the distributive property applies. Multiply each term within the parentheses by these factors as follows:</p>
<p class="para block" id="fwk-redden-ch01_s06_s02_p02"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1394" display="block"><mtable columnspacing="0.1em" columnalign="left"><mtr><mtd columnalign="left"><mo>+</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>+</mo><mn>1</mn><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>a</mi><mo>+</mo><mrow><mo>(</mo><mrow><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>b</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><mo>+</mo><mi>b</mi></mtd></mtr><mtr><mtd columnalign="left"><mo>−</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>a</mi><mo>+</mo><mrow><mo>(</mo><mrow><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mi>b</mi></mtd><mtd columnalign="left"><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mi>a</mi><mo>−</mo><mi>b</mi></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s06_s02_p03">Use this idea as a means to eliminate parentheses when adding and subtracting polynomials.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s02_n01">
<h3 class="title">Example 3</h3>
<p class="para" id="fwk-redden-ch01_s06_s02_p04">Add: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1395" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s02_p05">The property <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1396" display="inline"><mrow><mo>+</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi></mrow></math></span> allows us to eliminate the parentheses, after which we can then combine like terms.</p>
<p class="para" id="fwk-redden-ch01_s06_s02_p06"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1397" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s02_p07">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1398" display="inline"><mrow><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s06_s02_n02">
<h3 class="title">Example 4</h3>
<p class="para" id="fwk-redden-ch01_s06_s02_p08">Add: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1399" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><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>9</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s02_p09">Remember that the variable parts have to be exactly the same before we can add the coefficients.</p>
<p class="para" id="fwk-redden-ch01_s06_s02_p10"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1400" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="left"><mrow><mo>(</mo><mrow><mn>3</mn><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>9</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd><mo>=</mo><mstyle color="#007fbf"><munder accentunder="true"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow><mo stretchy="true">–</mo></munder></mstyle><mo>−</mo><mstyle color="#007f3f"><munder accentunder="true"><mrow><munder accentunder="true"><mrow><mn>4</mn><mi>x</mi><mi>y</mi></mrow><mo stretchy="true">–</mo></munder></mrow><mo stretchy="true">–</mo></munder></mstyle><mo>+</mo><munder accentunder="true"><mrow><munder accentunder="true"><mrow><munder accentunder="true"><mrow><mtext> </mtext><mn>9</mn><mtext> </mtext></mrow><mo stretchy="true">–</mo></munder></mrow><mo stretchy="true">–</mo></munder></mrow><mo stretchy="true">–</mo></munder><mo>+</mo><mstyle color="#007fbf"><munder accentunder="true"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow><mo stretchy="true">–</mo></munder></mstyle><mo>−</mo><mstyle color="#007f3f"><munder accentunder="true"><mrow><munder accentunder="true"><mrow><mn>6</mn><mi>x</mi><mi>y</mi></mrow><mo stretchy="true">–</mo></munder></mrow><mo stretchy="true">–</mo></munder></mstyle><mo>−</mo><munder accentunder="true"><mrow><munder accentunder="true"><mrow><munder accentunder="true"><mrow><mtext> </mtext><mn>7</mn><mtext> </mtext></mrow><mo stretchy="true">–</mo></munder></mrow><mo stretchy="true">–</mo></munder></mrow><mo stretchy="true">–</mo></munder></mtd></mtr><mtr><mtd columnalign="left"><mo>=</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s02_p11">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1401" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s02_p12">When subtracting polynomials, the parentheses become very important.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s02_n03">
<h3 class="title">Example 5</h3>
<p class="para" id="fwk-redden-ch01_s06_s02_p13">Subtract: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1402" display="inline"><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s02_p14">The property <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1403" display="inline"><mrow><mo>−</mo><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow><mo>=</mo><mo>−</mo><mi>a</mi><mo>−</mo><mi>b</mi></mrow></math></span> allows us to remove the parentheses after subtracting each term.</p>
<p class="para" id="fwk-redden-ch01_s06_s02_p15"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1404" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s02_p16">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1405" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s02_p17">Subtracting a quantity is equivalent to multiplying it by −1.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s02_n04">
<h3 class="title">Example 6</h3>
<p class="para" id="fwk-redden-ch01_s06_s02_p18">Subtract: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1406" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s02_p19">Distribute the −1, remove the parentheses, and then combine like terms. Multiplying the terms of a polynomial by −1 changes all the signs.</p>
<div class="informalfigure large">
<img src="section_04/19157971f267dfed73125fabab9d74ae.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s02_p21"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1407" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mi>y</mi><mo>−</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mi>y</mi><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s02_p22">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1408" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mi>y</mi><mo>−</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s06_s02_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch01_s06_s02_p23"><strong class="emphasis bold">Try this!</strong> Subtract: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1409" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>7</mn><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>5</mn><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s02_p24">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1410" display="inline"><mrow><mn>6</mn><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></math></span></p>
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</div>
</div>
</div>
<div class="section" id="fwk-redden-ch01_s06_s03" version="5.0" lang="en">
<h2 class="title editable block">Multiplying Polynomials</h2>
<p class="para block" id="fwk-redden-ch01_s06_s03_p01">Use the product rule for exponents, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1411" display="inline"><mrow><msup><mi>x</mi><mi>m</mi></msup><mo>⋅</mo><msup><mi>x</mi><mi>n</mi></msup><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow></math></span>, to multiply a monomial times a polynomial. In other words, when multiplying two expressions with the same base, add the exponents. To find the product of monomials, multiply the coefficients and add the exponents of variable factors with the same base. For example,</p>
<p class="para block" id="fwk-redden-ch01_s06_s03_p02"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1412" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mn>7</mn><msup><mi>x</mi><mn>4</mn></msup><mo>⋅</mo><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>7</mn><mo>⋅</mo><mn>8</mn><mo>⋅</mo><msup><mi>x</mi><mn>4</mn></msup><mo>⋅</mo><msup><mi>x</mi><mn>3</mn></msup></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>e</mi><mtext> </mtext><mi>p</mi><mi>r</mi><mi>o</mi><mi>p</mi><mi>e</mi><mi>r</mi><mi>t</mi><mi>y</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>56</mn><msup><mi>x</mi><mrow><mn>4</mn><mo>+</mo><mn>3</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mi>P</mi><mi>r</mi><mi>o</mi><mi>d</mi><mi>u</mi><mi>c</mi><mi>t</mi><mtext> </mtext><mi>r</mi><mi>u</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>x</mi><mi>p</mi><mi>o</mi><mi>n</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>56</mn><msup><mi>x</mi><mn>7</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s06_s03_p03">To multiply a polynomial by a monomial, apply the distributive property, and then simplify each term.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s03_n01">
<h3 class="title">Example 7</h3>
<p class="para" id="fwk-redden-ch01_s06_s03_p04">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1413" display="inline"><mrow><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>2</mn><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><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s03_p05">Apply the distributive property and then simplify.</p>
<div class="informalfigure large">
<img src="section_04/2603f8e94a90164b1620a009c51f7eaf.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s03_p07"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1414" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mstyle><mo>⋅</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mstyle color="#007fbf"><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mstyle><mo>⋅</mo><mi>x</mi><mi>y</mi><mo>+</mo><mstyle color="#007fbf"><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mstyle><mo>⋅</mo><mn>1</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>10</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>4</mn></msup><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s03_p08">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1415" display="inline"><mrow><mn>10</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>4</mn></msup><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s03_p09">To summarize, multiplying a polynomial by a monomial involves the distributive property and the product rule for exponents. Multiply all of the terms of the polynomial by the monomial. For each term, multiply the coefficients and add exponents of variables where the bases are the same.</p>
<p class="para block" id="fwk-redden-ch01_s06_s03_p10">In the same manner that we used the distributive property to distribute a monomial, we use it to distribute a binomial.
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1416" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mstyle color="#007fbf"><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mstyle></mrow><mrow><mo>(</mo><mrow><mi>c</mi><mo>+</mo><mi>d</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mstyle></mrow><mo>⋅</mo><mi>c</mi><mo>+</mo><mrow><mstyle color="#007fbf"><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mstyle></mrow><mo>⋅</mo><mi>d</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>d</mi></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mi>a</mi><mi>c</mi><mo>+</mo><mi>a</mi><mi>d</mi><mo>+</mo><mi>b</mi><mi>c</mi><mo>+</mo><mi>b</mi><mi>d</mi></mtd></mtr></mtable></math></span>
Here we apply the distributive property multiple times to produce the final result. This same result is obtained in one step if we apply the distributive property to <em class="emphasis">a</em> and <em class="emphasis">b</em> separately as follows:</p>
<div class="informalfigure large block">
<img src="section_04/7be734ec5c654af428357e9bd13b2a08.png">
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s03_p14">This is often called the FOIL method. Multiply the first, outer, inner, and then last terms.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s03_n02">
<h3 class="title">Example 8</h3>
<p class="para" id="fwk-redden-ch01_s06_s03_p15">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1417" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s03_p16">Distribute 6<em class="emphasis">x</em> and −1 and then combine like terms.</p>
<p class="para" id="fwk-redden-ch01_s06_s03_p17"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1418" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>6</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mstyle color="#007fbf"><mn>6</mn><mi>x</mi></mstyle><mo>⋅</mo><mn>3</mn><mi>x</mi><mo>−</mo><mstyle color="#007fbf"><mn>6</mn><mi>x</mi></mstyle><mo>⋅</mo><mn>5</mn><mo>+</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>3</mn><mi>x</mi><mo>−</mo><mrow><mo>(</mo><mrow><mstyle color="#007f3f"><mo>−</mo><mn>1</mn></mstyle></mrow><mo>)</mo></mrow><mo>⋅</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>30</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>33</mn><mi>x</mi><mo>+</mo><mn>5</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s03_p18">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1419" display="inline"><mrow><mn>18</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>33</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s03_p19">Consider the following two calculations:</p>
<p class="para block" id="fwk-redden-ch01_s06_s03_p20">
</p>
<div class="informaltable"> <table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1420" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>a</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1421" display="inline"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><msup><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>−</mo><mi>b</mi><mi>a</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>−</mo><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para block" id="fwk-redden-ch01_s06_s03_p21">This leads us to two formulas that describe <span class="margin_term"><a class="glossterm">perfect square trinomials</a><span class="glossdef">The trinomials obtained by squaring the binomials <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1422" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1423" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></span></span>:</p>
<p class="para block" id="fwk-redden-ch01_s06_s03_p22"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1424" display="block"><mtable columnspacing="0.1em"><mtr><mtd><msup><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><msup><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para editable block" id="fwk-redden-ch01_s06_s03_p23">We can use these formulas to quickly square a binomial.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s03_n03">
<h3 class="title">Example 9</h3>
<p class="para" id="fwk-redden-ch01_s06_s03_p24">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1425" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s03_p25">Here <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1426" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>3</mn><mi>x</mi></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1427" display="inline"><mrow><mi>b</mi><mo>=</mo><mn>5</mn></mrow><mo>.</mo></math></span> Apply the formula:</p>
<div class="informalfigure large">
<img src="section_04/6f9efdeb4dd8ce181d4304e3f8ada495.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s03_p27">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1428" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch01_s06_s03_p28">This process should become routine enough to be performed mentally. Our third special product follows:
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1429" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>a</mi><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mstyle color="#ff0000"><mo>−</mo><mi>a</mi><mi>b</mi><mo>+</mo><mi>a</mi><mi>b</mi></mstyle><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span>
This product is called <span class="margin_term"><a class="glossterm">difference of squares</a><span class="glossdef">The special product obtained by multiplying conjugate binomials <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1430" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span></span></span>:</p>
<p class="para block" id="fwk-redden-ch01_s06_s03_p31"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1431" display="block"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></math></span></p>
<p class="para block" id="fwk-redden-ch01_s06_s03_p32">The binomials <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1432" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1433" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span> are called <span class="margin_term"><a class="glossterm">conjugate binomials</a><span class="glossdef">The binomials <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1434" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow></math></span> and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1435" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></span></span>. When multiplying conjugate binomials the middle terms are opposites and their sum is zero; the product is itself a binomial.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s03_n04">
<h3 class="title">Example 10</h3>
<p class="para" id="fwk-redden-ch01_s06_s03_p33">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1436" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s03_p34"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1437" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><msup><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>−</mo><msup><mn>1</mn><mn>2</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s03_p35">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1438" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s06_s03_n04a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch01_s06_s03_p36"><strong class="emphasis bold">Try this!</strong> Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1439" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><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>5</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s03_p38">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1440" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>25</mn><msup><mi>y</mi><mn>4</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
<div class="mediaobject">
<a data-iframe-code='<iframe src="http://www.youtube.com/v/p7R3FdPp6_s" condition="http://img.youtube.com/vi/p7R3FdPp6_s/0.jpg" vendor="youtube" width="450" height="340" scalefit="1"></iframe>' href="http://www.youtube.com/v/p7R3FdPp6_s" class="replaced-iframe" onclick="return replaceIframe(this)">(click to see video)</a>
</div>
</div>
<div class="callout block" id="fwk-redden-ch01_s06_s03_n05">
<h3 class="title">Example 11</h3>
<p class="para" id="fwk-redden-ch01_s06_s03_p39">Multiply: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1441" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s03_p40">Here we perform one product at a time.</p>
<div class="informalfigure large">
<img src="section_04/94bf5b5472fe2ef54e090df103f39d85.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s03_p42">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1442" display="inline"><mrow><mn>125</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>150</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>60</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow></math></span></p>
</div>
</div>
<div class="section" id="fwk-redden-ch01_s06_s04" version="5.0" lang="en">
<h2 class="title editable block">Dividing Polynomials</h2>
<p class="para block" id="fwk-redden-ch01_s06_s04_p01">Use the quotient rule for exponents, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1443" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mi>m</mi></msup></mrow><mrow><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mo>=</mo><msup><mi>x</mi><mrow><mi>m</mi><mo>−</mo><mi>n</mi></mrow></msup></mrow></math></span>, to divide a polynomial by a monomial. In other words, when dividing two expressions with the same base, subtract the exponents. In this section, we will assume that all variables in the denominator are nonzero.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s04_n01">
<h3 class="title">Example 12</h3>
<p class="para" id="fwk-redden-ch01_s06_s04_p02">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1444" display="inline"><mrow><mfrac><mrow><mn>24</mn><msup><mi>x</mi><mn>7</mn></msup><msup><mi>y</mi><mn>5</mn></msup></mrow><mrow><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p03">Divide the coefficients and apply the quotient rule by subtracting the exponents of the like bases.</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p04"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1445" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mn>24</mn><msup><mi>x</mi><mn>7</mn></msup><msup><mi>y</mi><mn>5</mn></msup></mrow><mrow><mn>8</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mfrac><mrow><mn>24</mn></mrow><mn>8</mn></mfrac><msup><mi>x</mi><mrow><mn>7</mn><mo>−</mo><mn>3</mn></mrow></msup><msup><mi>y</mi><mrow><mn>5</mn><mo>−</mo><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p05">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1446" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><msup><mi>y</mi><mn>3</mn></msup></mrow></math></span></p>
</div>
<p class="para block" id="fwk-redden-ch01_s06_s04_p06">When dividing a polynomial by a monomial, we may treat the monomial as a common denominator and break up the fraction using the following property:
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1447" display="block"><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mi>c</mi></mfrac><mo>=</mo><mfrac><mi>a</mi><mi>c</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mi>c</mi></mfrac></mrow></math></span>
Applying this property will result in terms that can be treated as quotients of monomials.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s04_n02">
<h3 class="title">Example 13</h3>
<p class="para" id="fwk-redden-ch01_s06_s04_p09">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1448" display="inline"><mrow><mfrac><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>25</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>15</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p10">Break up the fraction by dividing each term in the numerator by the monomial in the denominator, and then simplify each term.</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p11"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1449" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mfrac><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>25</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>15</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>25</mn><msup><mi>x</mi><mn>3</mn></msup></mrow><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>15</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mfrac><mn>5</mn><mn>5</mn></mfrac><msup><mi>x</mi><mrow><mn>4</mn><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mn>25</mn></mrow><mn>5</mn></mfrac><msup><mi>x</mi><mrow><mn>3</mn><mo>−</mo><mn>2</mn></mrow></msup><mo>−</mo><mfrac><mrow><mn>15</mn></mrow><mn>5</mn></mfrac><msup><mi>x</mi><mrow><mn>2</mn><mo>−</mo><mn>2</mn></mrow></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mn>1</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>0</mn></msup></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo>⋅</mo><mn>1</mn></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p12">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1450" display="inline"><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s04_p13">We can check our division by multiplying our answer, the quotient, by the monomial in the denominator, the divisor, to see if we obtain the original numerator, the dividend.</p>
<div class="informaltable block">
<table cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1451" display="inline"><mstyle color="#007fbf"><mfrac><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>d</mi></mrow><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>o</mi><mi>r</mi></mrow></mfrac><mo>=</mo><mi>Q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi></mstyle></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1452" display="inline"><mrow><mfrac><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>25</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>15</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p></td>
</tr>
<tr>
<td align="center"><p class="para">or</p></td>
<td align="center"><p class="para">or</p></td>
</tr>
<tr>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1453" display="inline"><mrow><mstyle color="#007fbf"><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>d</mi><mo>=</mo><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>o</mi><mi>r</mi><mo>⋅</mo><mi>Q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi></mstyle></mrow></math></span></p></td>
<td align="center"><p class="para"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1454" display="inline"><mrow><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>25</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>15</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mrow></math></span></p></td>
</tr>
</tbody>
</table>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s04_p15">The same technique outlined for dividing by a monomial <em class="emphasis bolditalic">does not</em> work for polynomials with two or more terms in the denominator. In this section, we will outline a process called <span class="margin_term"><a class="glossterm">polynomial long division</a><span class="glossdef">The process of dividing two polynomials using the division algorithm.</span></span>, which is based on the division algorithm for real numbers. For the sake of clarity, we will assume that all expressions in the denominator are nonzero.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s04_n03">
<h3 class="title">Example 14</h3>
<p class="para" id="fwk-redden-ch01_s06_s04_p16">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1455" display="inline"><mrow><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p17">Here <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1456" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span> is the divisor and <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1457" display="inline"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow></math></span> is the dividend. To determine the first term of the quotient, divide the leading term of the dividend by the leading term of the divisor.</p>
<div class="informalfigure large">
<img src="section_04/597c1846212bb42a41ee60d1ea564951.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p19">Multiply the first term of the quotient by the divisor, remembering to distribute, and line up like terms with the dividend.</p>
<div class="informalfigure large">
<img src="section_04/5d4e3953ce01091edb8660ea2cbbcd05.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p21">Subtract the resulting quantity from the dividend. Take care to subtract both terms.</p>
<div class="informalfigure large">
<img src="section_04/59834d557ea8a06825dbda67dbb2956b.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p23">Bring down the remaining terms and repeat the process.</p>
<div class="informalfigure large">
<img src="section_04/8e8f8648f001e0f2ebe3d06303378e50.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p25">Notice that the leading term is eliminated and that the result has a degree that is one less. The complete process is illustrated below:</p>
<div class="informalfigure large">
<img src="section_04/12ac7ce3d68c9c0bc218027f83ce92f0.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p27">Polynomial long division ends when the degree of the remainder is less than the degree of the divisor. Here, the remainder is 0. Therefore, the binomial divides the polynomial evenly and the answer is the quotient shown above the division bar.</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p28"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1458" display="block"><mrow><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>4</mn></mrow><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p29">To check the answer, multiply the divisor by the quotient to see if you obtain the dividend as illustrated below:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p30"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1459" display="block"><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>8</mn><mi>x</mi><mo>−</mo><mn>4</mn><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p31">This is left to the reader as an exercise.</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p32">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1460" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s04_p33">Next, we demonstrate the case where there is a nonzero remainder.</p>
<div class="informalfigure large block">
<img src="section_04/e7aecc5db5f3a690ad915a7012bd1474.png">
</div>
<p class="para block" id="fwk-redden-ch01_s06_s04_p35">Just as with real numbers, the final answer adds to the quotient the fraction where the remainder is the numerator and the divisor is the denominator. In general, when dividing we have:
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1461" display="block"><mrow><mfrac><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>d</mi></mrow><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>o</mi><mi>r</mi></mrow></mfrac><mo>=</mo><mstyle color="#007fbf"><mi>Q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi></mstyle><mo>+</mo><mfrac><mrow><mstyle color="#007f3f"><mi>R</mi><mi>e</mi><mi>m</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>r</mi></mstyle></mrow><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>o</mi><mi>r</mi></mrow></mfrac></mrow></math></span>
If we multiply both sides by the divisor we obtain,
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1462" display="block"><mrow><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>d</mi><mo>=</mo><mstyle color="#007fbf"><mi>Q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi></mstyle><mo>×</mo><mi>D</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>o</mi><mi>r</mi><mo>+</mo><mstyle color="#007f3f"><mi>R</mi><mi>e</mi><mi>m</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>r</mi></mstyle></mrow></math></span></p>
<div class="callout block" id="fwk-redden-ch01_s06_s04_n04">
<h3 class="title">Example 15</h3>
<p class="para" id="fwk-redden-ch01_s06_s04_p39">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1463" display="inline"><mrow><mfrac><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p40">Since the denominator is a binomial, begin by setting up polynomial long division.</p>
<div class="informalfigure large">
<img src="section_04/907e3b78b0543ab03b8e48c0db9fc92a.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p42">To start, determine what monomial times <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1464" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span> results in a leading term <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1465" display="inline"><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>.</mo></math></span> This is the quotient of the given leading terms: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1466" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>÷</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi></mrow><mo>)</mo></mrow><mo>=</mo><mn>3</mn><mi>x</mi></mrow><mo>.</mo></math></span> Multiply <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1467" display="inline"><mrow><mn>3</mn><mi>x</mi></mrow></math></span> times the divisor <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1468" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span>, and line up the result with like terms of the dividend.</p>
<div class="informalfigure large">
<img src="section_04/0bf0959dce4e14a83cd7c030a7415b9d.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p44">Subtract the result from the dividend and bring down the constant term +3.</p>
<div class="informalfigure large">
<img src="section_04/6637d797c48b15341303c31af6093ade.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p46">Subtracting eliminates the leading term. Multiply <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1469" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span> by −1 and line up the result.</p>
<div class="informalfigure large">
<img src="section_04/e197703db2295db06f984ba307bcf5c2.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p48">Subtract again and notice that we are left with a remainder.</p>
<div class="informalfigure large">
<img src="section_04/da6de19c485bfb98d24aee7dbea7dae9.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p50">The constant term 2 has degree 0 and thus the division ends. Therefore,</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p51"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1470" display="block"><mrow><mfrac><mrow><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mstyle color="#007fbf"><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mstyle><mo>+</mo><mfrac><mstyle color="#007f3f"><mn>2</mn></mstyle><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p52">To check that this result is correct, we multiply as follows:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p53"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1471" display="block"><mtable columnspacing="0.1em"><mtr><mtd columnalign="right"><mstyle color="#007fbf"><mi>q</mi><mi>u</mi><mi>o</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>n</mi><mi>t</mi></mstyle><mo>×</mo><mi>d</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>o</mi><mi>r</mi><mo>+</mo><mstyle color="#007f3f"><mi>r</mi><mi>e</mi><mi>m</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>r</mi></mstyle></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mrow><mstyle color="#007fbf"><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mstyle></mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007f3f"><mn>2</mn></mstyle></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mo>+</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>2</mn></mtd></mtr><mtr><mtd columnalign="right"></mtd><mtd><mo>=</mo></mtd><mtd columnalign="left"><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mtext> </mtext><mo>=</mo><mtext> </mtext><mi>d</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mstyle color="#007fbf"><mtext>✓</mtext></mstyle></mtd></mtr></mtable></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p54">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1472" display="inline"><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>+</mo><mfrac><mn>2</mn><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s04_p55">Occasionally, some of the powers of the variables appear to be missing within a polynomial. This can lead to errors when lining up like terms. Therefore, when first learning how to divide polynomials using long division, fill in the missing terms with zero coefficients, called <span class="margin_term"><a class="glossterm">placeholders</a><span class="glossdef">Terms with zero coefficients used to fill in all missing exponents within a polynomial.</span></span>.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s04_n05">
<h3 class="title">Example 16</h3>
<p class="para" id="fwk-redden-ch01_s06_s04_p56">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1473" display="inline"><mrow><mfrac><mrow><mn>27</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>64</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p57">Notice that the binomial in the numerator does not have terms with degree 2 or 1. The division is simplified if we rewrite the expression with placeholders:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p58"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1474" display="block"><mrow><mn>27</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>64</mn><mo>=</mo><mn>27</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mstyle color="#007f3f"><mn>0</mn><msup><mi>x</mi><mn>2</mn></msup></mstyle><mo>+</mo><mstyle color="#007f3f"><mn>0</mn><mi>x</mi></mstyle><mo>+</mo><mn>64</mn></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p59">Set up polynomial long division:</p>
<div class="informalfigure large">
<img src="section_04/3580ae29e84dc2142822da2ab8c2e345.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p61">We begin with <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1475" display="inline"><mrow><mn>27</mn><msup><mi>x</mi><mn>3</mn></msup><mo>÷</mo><mn>3</mn><mi>x</mi><mo>=</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span> and work the rest of the division algorithm.</p>
<div class="informalfigure large">
<img src="section_04/c1b11a0e625faf8dfcc54428eb1b21e5.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p63">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1476" display="inline"><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>12</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></math></span></p>
</div>
<div class="callout block" id="fwk-redden-ch01_s06_s04_n06">
<h3 class="title">Example 17</h3>
<p class="para" id="fwk-redden-ch01_s06_s04_p64">Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1477" display="inline"><mrow><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>23</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="simpara">Solution:</p>
<div class="informalfigure large">
<img src="section_04/179294b59c7365cd517ffacf1e4e7086.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p66">Begin the process by dividing the leading terms to determine the leading term of the quotient <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1478" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><mo>÷</mo><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mstyle color="#007fbf"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mstyle></mrow><mo>.</mo></math></span> Take care to distribute and line up the like terms. Continue the process until the remainder has a degree less than 2.</p>
<div class="informalfigure large">
<img src="section_04/6712e898302af732f5e899c4fc6abc3b.png">
</div>
<p class="para" id="fwk-redden-ch01_s06_s04_p68">The remainder is <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1479" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>.</mo></math></span> Write the answer with the remainder:</p>
<p class="para" id="fwk-redden-ch01_s06_s04_p69"><span class="informalequation"><math xml:id="fwk-redden-ch01_m1480" display="block"><mrow><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>23</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac></mrow></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p70">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1481" display="inline"><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfrac></mrow></math></span></p>
</div>
<p class="para editable block" id="fwk-redden-ch01_s06_s04_p71">Polynomial long division takes time and practice to master. Work lots of problems and remember that you may check your answers by multiplying the quotient by the divisor (and adding the remainder if present) to obtain the dividend.</p>
<div class="callout block" id="fwk-redden-ch01_s06_s04_n06a">
<h3 class="title"></h3>
<p class="para" id="fwk-redden-ch01_s06_s04_p72"><strong class="emphasis bold">Try this!</strong> Divide: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1482" display="inline"><mrow><mfrac><mrow><mn>6</mn><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>13</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>14</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi> </mi></mrow><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow><mo>.</mo></math></span></p>
<p class="para" id="fwk-redden-ch01_s06_s04_p73">Answer: <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1483" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>−</mo><mn>4</mn><mo>−</mo><mfrac><mn>2</mn><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span></p>
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</div>
<div class="key_takeaways editable block" id="fwk-redden-ch01_s06_s04_n07">
<h3 class="title">Key Takeaways</h3>
<ul class="itemizedlist" id="fwk-redden-ch01_s06_s04_l01" mark="bullet">
<li>Polynomials are special algebraic expressions where the terms are the products of real numbers and variables with whole number exponents.</li>
<li>The degree of a polynomial with one variable is the largest exponent of the variable found in any term. In addition, the terms of a polynomial are typically arranged in descending order based on the degree of each term.</li>
<li>When adding polynomials, remove the associated parentheses and then combine like terms. When subtracting polynomials, distribute the −1, remove the parentheses, and then combine like terms.</li>
<li>To multiply polynomials apply the distributive property; multiply each term in the first polynomial with each term in the second polynomial. Then combine like terms.</li>
<li>When dividing by a monomial, divide all terms in the numerator by the monomial and then simplify each term.</li>
<li>When dividing a polynomial by another polynomial, apply the division algorithm.</li>
</ul>
</div>
<div class="qandaset block" id="fwk-redden-ch01_s06_qs01" defaultlabel="number">
<h3 class="title">Topic Exercises</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s06_qs01_qd01">
<h3 class="title">Part A: Definitions</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s06_qs01_qd01_qd01">
<p class="para" id="fwk-redden-ch01_s06_qs01_p01"><strong class="emphasis bold">Write the given polynomials in standard form.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa01">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p02"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1484" display="inline"><mrow><mn>1</mn><mo>−</mo><mi>x</mi><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa02">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p04"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1486" display="inline"><mrow><mi>y</mi><mo>−</mo><mn>5</mn><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa03">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p06"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1488" display="inline"><mrow><mi>y</mi><mo>−</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mo>−</mo><msup><mi>y</mi><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa04">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p08"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1490" display="inline"><mrow><mn>8</mn><mo>−</mo><mn>12</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>3</mn></msup><mo>−</mo><mi>a</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa05">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p10"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1492" display="inline"><mrow><mn>2</mn><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa06">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p12"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1494" display="inline"><mrow><msup><mi>a</mi><mn>3</mn></msup><mo>−</mo><mn>5</mn><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>a</mi><mn>4</mn></msup><mo>−</mo><msup><mi>a</mi><mn>5</mn></msup><mo>+</mo><mn>6</mn><mi>a</mi></mrow></math></span></p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s06_qs01_qd01_qd02" start="7">
<p class="para" id="fwk-redden-ch01_s06_qs01_p14"><strong class="emphasis bold">Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa07">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p15"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1496" 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-ch01_s06_qs01_qa08">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p17"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1497" display="inline"><mrow><mn>5</mn><mo>−</mo><mn>10</mn><msup><mi>x</mi><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa09">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p19"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1498" 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>6</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa10">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p21"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1499" display="inline"><mrow><mo>−</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa11">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p23"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1500" display="inline"><mrow><msup><mi>x</mi><mn>4</mn></msup><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa12">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p25">5</p>
</div>
</li>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s06_qs01_qd01_qd03" start="13">
<p class="para" id="fwk-redden-ch01_s06_qs01_p27"><strong class="emphasis bold">State whether the polynomial is linear or quadratic and give the leading coefficient.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa13">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p28"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1501" display="inline"><mrow><mn>1</mn><mo>−</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa14">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p30"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1502" display="inline"><mrow><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa15">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p32"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1503" display="inline"><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa16">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p34"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1504" display="inline"><mrow><mn>100</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa17">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p36"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1505" display="inline"><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa18">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p38"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1506" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa19">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p40"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1507" display="inline"><mrow><mi>x</mi><mo>−</mo><mn>6</mn><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-ch01_s06_qs01_qa20">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p42"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1508" display="inline"><mrow><mn>1</mn><mo>−</mo><mn>5</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s06_qs01_qd02">
<h3 class="title">Part B: Adding and Subtracting Polynomials</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s06_qs01_qd02_qd01" start="21">
<p class="para" id="fwk-redden-ch01_s06_qs01_p44"><strong class="emphasis bold">Simplify.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa21">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p45"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1509" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa22">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p47"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1511" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>−</mo><mn>12</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa23">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p49"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1513" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>10</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>10</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa24">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p51"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1515" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa25">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p53"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1517" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa26">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p55"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1519" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa27">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p57"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1521" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>2</mn></mfrac><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa28">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p59"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1523" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>4</mn><mn>5</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>5</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mn>10</mn></mrow><mn>6</mn></mfrac></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>3</mn><mrow><mn>10</mn></mrow></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>5</mn></mfrac></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa29">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p61"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1525" display="inline"><mrow><mrow><mo>(</mo><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>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>2</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>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa30">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p63"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1527" 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><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa31">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p65"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1529" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>a</mi><mi>b</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>7</mn><mi>a</mi><mi>b</mi><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>4</mn><mo>−</mo><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa32">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p67"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1531" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mi>a</mi><mi>b</mi><mo>−</mo><mn>6</mn><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><mn>7</mn><mi>a</mi><mi>b</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa33">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p69"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1533" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>−</mo><mn>8</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>5</mn><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>−</mo><mn>4</mn><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa34">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p71"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1535" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>m</mi><mn>2</mn></msup><mi>n</mi><mo>−</mo><mn>6</mn><mi>m</mi><mi>n</mi><mo>+</mo><mn>9</mn><mi>m</mi><msup><mi>n</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>m</mi><mn>2</mn></msup><mi>n</mi><mo>+</mo><mn>10</mn><mi>m</mi><mi>n</mi></mrow><mo>)</mo></mrow><mo>−</mo><msup><mi>m</mi><mn>2</mn></msup><mi>n</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa35">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p73"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1537" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>8</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>2</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mi>y</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa36">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p75"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1539" 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><mo>−</mo><mrow><mo>(</mo><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>x</mi><mi>y</mi><mo>−</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa37">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p77"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1541" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>3</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>4</mn><mn>5</mn></mfrac><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><mn>11</mn></mrow><mn>8</mn></mfrac><mi>a</mi><mi>b</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa38">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p79"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1543" display="inline"><mrow><mrow><mo>(</mo><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><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><mo>−</mo><mrow><mo>(</mo><mrow><mfrac><mn>7</mn><mn>5</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mi>y</mi><mo>+</mo><mfrac><mn>7</mn><mn>3</mn></mfrac><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>x</mi><mi>y</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa39">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p81"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1545" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>5</mn><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa40">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p83"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1547" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>7</mn><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><mn>5</mn></mrow><mo>)</mo></mrow><mo>−</mo><mrow><mo>(</mo><mrow><mn>6</mn><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><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa41">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p85">Subtract <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1549" display="inline"><mrow><mn>4</mn><mi>y</mi><mo>−</mo><mn>3</mn></mrow></math></span> from <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1550" display="inline"><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>y</mi><mo>−</mo><mn>10</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa42">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p87">Subtract <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1552" display="inline"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>2</mn></mrow></math></span> from <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1553" display="inline"><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>.</mo></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa43">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p89">A right circular cylinder has a height that is equal to the radius of the base, <span class="inlineequation"><math xml:id="fwk-redden-ch01_m1555" display="inline"><mrow><mi>h</mi><mo>=</mo><mi>r</mi></mrow><mo>.</mo></math></span> Find a formula for the surface area in terms of <em class="emphasis">h</em>.</p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa44">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p91">A rectangular solid has a width that is twice the height and a length that is 3 times that of the height. Find a formula for the surface area in terms of the height.</p>
</div>
</li>
</ol>
</ol>
<ol class="qandadiv" id="fwk-redden-ch01_s06_qs01_qd03">
<h3 class="title">Part C: Multiplying Polynomials</h3>
<ol class="qandadiv" id="fwk-redden-ch01_s06_qs01_qd03_qd01" start="45">
<p class="para" id="fwk-redden-ch01_s06_qs01_p93"><strong class="emphasis bold">Multiply.</strong></p>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa45">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p94"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1558" display="inline"><mrow><mo>−</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>⋅</mo><mn>2</mn><mi>x</mi></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa46">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p96"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1560" display="inline"><mrow><mo>−</mo><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mi>y</mi><mo>⋅</mo><mn>5</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>y</mi><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa47">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p98"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1562" display="inline"><mrow><mn>2</mn><mi>x</mi><mrow><mo>(</mo><mrow><mn>5</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa48">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p100"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1564" display="inline"><mrow><mo>−</mo><mn>4</mn><mi>x</mi><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa49">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p102"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1566" display="inline"><mrow><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa50">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p104"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1568" 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><mi>x</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa51">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p106"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1570" display="inline"><mrow><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>4</mn></msup><mrow><mo>(</mo><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa52">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p108"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1572" display="inline"><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><msup><mi>a</mi><mn>3</mn></msup><mrow><mo>(</mo><mrow><mn>24</mn><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>a</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa53">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p110"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1574" display="inline"><mrow><mn>2</mn><mi>x</mi><mi>y</mi><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mi>x</mi><mi>y</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa54">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1576" display="block"><mrow><mo>−</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>a</mi><mi>b</mi><mo>+</mo><mn>5</mn><msup><mi>b</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa55">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p114"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1578" display="inline"><mrow><msup><mi>x</mi><mi>n</mi></msup><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa56">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p116"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1580" display="inline"><mrow><msup><mi>x</mi><mi>n</mi></msup><mrow><mo>(</mo><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><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa57">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p118"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1582" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>4</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa58">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p120"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1584" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa59">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p122"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1586" 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>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa60">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p124"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1588" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>9</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa61">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p126"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1590" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</mn><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>5</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-ch01_s06_qs01_qa62">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p128"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1592" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa63">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p130"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1594" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>3</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>5</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa64">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p132"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1596" 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>6</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa65">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p134"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1598" display="inline"><mrow><mrow><mo>(</mo><mrow><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><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>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa66">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p136"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1600" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mi>b</mi><mo>+</mo><mn>7</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>a</mi><mi>b</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa67">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p138"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1602" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>4</mn><mi>x</mi><mo>−</mo><mn>5</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa68">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p140"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1604" 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><mo>−</mo><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa69">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p142"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1606" display="inline"><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa70">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p144"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1608" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>7</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa71">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p146"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1610" display="inline"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>7</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa72">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p148"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1612" display="inline"><mrow><mrow><mo>(</mo><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mi>x</mi><mo>+</mo><mn>6</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa73">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p150"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1614" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>8</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa74">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p152"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1616" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa75">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p154"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1618" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa76">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p156"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1620" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa77">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p158"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1622" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>−</mo><mn>3</mn><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa78">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p160"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1624" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>7</mn><mi>a</mi><mo>−</mo><mi>b</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa79">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1626" display="block"><mrow><msup><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><mn>2</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa80">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1628" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>6</mn><mi>y</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa81">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1630" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>−</mo><mi>a</mi><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa82">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1632" display="block"><mrow><msup><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa83">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p170"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1634" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa84">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p172"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1636" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa85">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p174"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1638" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa86">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p176"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1640" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa87">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p178"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1642" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa88">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p180"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1644" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa89">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p182"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1646" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa90">
<div class="question">
<p class="para" id="fwk-redden-ch01_s06_qs01_p184"><span class="inlineequation"><math xml:id="fwk-redden-ch01_m1648" display="inline"><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><mi>x</mi><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mn>4</mn></msup></mrow></math></span></p>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa91">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1650" display="block"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>+</mo><mn>5</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>−</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>
<li class="qandaentry" id="fwk-redden-ch01_s06_qs01_qa92">
<div class="question">
<span class="informalequation"><math xml:id="fwk-redden-ch01_m1652" display="block"><mrow><mrow><mo>(</mo><mrow><msup><mi>x</mi><mi>n</mi></msup><mo>−</mo><mn>1</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><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>3</mn></mrow><mo>)</mo></mrow></mrow></math></span>
</div>
</li>