1e4585d9-d764-49ac-ba6e-9b9ad0f33ecf.html

<p>In this unit, remember to double click on mathematical expressions/equations to enlarge, if needed.</p>
<h4>Dividing Polynomial Functions</h4>
<p>Just as polynomials can be divided, polynomial functions can also be divided.</p>
<br>
<div>
  <p><strong> DIVISION OF POLYNOMIAL FUNCTIONS </strong></p>
  <hr>
  <p>For functions \(f(x)\) and \(g(x)\), where \(g(x) \neq 0\),</p>
  <p>\((\frac{f}{g})(x)=\frac{f(x)}{g(x)}\)</p>
</div>
<br>
<p class="os-raise-text-bold">Example 1</p>
<p>For functions \(f(x)=x^2-5x-14\) and \(g(x)=x+2\):</p>
<ol class="os-raise-noindent" type="a">
  <li>Find \((\frac{f}{g})(x)\).</li>
</ol>
<p><strong>Step 1 -</strong> \((\frac{f}{g})(x)=\frac{f(x)}{g(x)}\)  Substitute for \(f(x)\) and \(g(x)\).</p>
<p> \((\frac{f}{g})(x)=\frac{f(x)=x^2-5x-14}{x+2}\)</p>
<p><strong>Step 2 -</strong> Divide the polynomials using long division.</p>
<p> \[\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x - 7\\x + 2\left){\vphantom{1{{x^2} - 5x - 14}}}\right.<br>
  \!\!\!\!\overline{\,\,\,\vphantom 1{{{x^2} - 5x - 14}}}\\\,\,\,\,\,\,\,\,\,\,\underline { - ({x^2} + 2x)} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 7x - 14\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - ( - 7x - 14)} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array}\]<br>
  \((\frac{f}{g})(x)=x-7\)</p>
<br>
<ol class="os-raise-noindent" start="2" type="a">
  <li>Find \((\frac{f}{g})(-4)\).</li>
</ol>
<p><strong>Step 1 -</strong> In part (1), we found \((\frac{f}{g})(x)\).</p>
<p> \((\frac{f}{g})(x)=x&minus;7\)</p>
<p><strong>Step 2 - </strong>To find \((\frac{f}{g})(-4)\), substitute \(x=-4\).</p>
<p> \((\frac{f}{g})(-4)=-4-7\)<br>
  \((\frac{f}{g})(-4)=-11\)</p>
<br>
<h4>Try It: Dividing Polynomial Functions</h4>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
  <div class="os-raise-ib-cta-content">
    <p>For functions \(f(x)=x^2-5x-36\) and \(g(x)=x+4\):</p>
    <ol class="os-raise-noindent">
      <li>Find \((\frac{f}{g})(x)\). </li>
    </ol>
    <ol class="os-raise-noindent" start="2">
      <li>Find \((\frac{f}{g})(-5)\). </li>
    </ol>
  </div>
  <div class="os-raise-ib-cta-prompt">
    <p>Write down your answers. Then select the <strong>solution</strong> button to compare your work. </p>
  </div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
  <p>Here is how to solve these polynomial function division problems:</p>
  <ol class="os-raise-noindent">
    <li>\((\frac{f}{g})(x) = \frac{x^2-5x-36}{x+4}=x-9\) </li>
  </ol>
  <ol class="os-raise-noindent" start="2">
    <li>\((\frac{f}{g})(-5)=x-9=(-5)-9=-14\) </li>
  </ol>
</div>
<br>
<h4>Using the Remainder Theorem</h4>
<p>Let&rsquo;s look at some division problems and their remainders. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as \(x - c\), the value of the function at \(c\), \(f(c)\), is the same as the remainder from the division problem.</p>
<table class="os-raise-wideadjustedtable">
  <thead>
    <tr>
      <th scope="col">Dividend</th>
      <th scope="col">Divisor</th>
      <th scope="col">Remainder</th>
      <th scope="col">Function</th>
      <th scope="col">\(f(c)\)</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>
        <p>\(x^4-7x^2+7x+6\)</p>
      </td>
      <td>
        <p>\(x - (-3)\)</p>
      </td>
      <td>
        <p>\(3\)</p>
      </td>
      <td>
        <p>\(f(x)=x^4-7x^2+7x+6\)</p>
      </td>
      <td>
        <p>\(f(-3)=3\)</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>\(f(x)=3x^3-2x^2-10x+8\)</p>
      </td>
      <td>
        <p>\(x - (-3)\)</p>
      </td>
      <td>
        <p>\(-61\)</p>
      </td>
      <td>
        <p>\(f(x)=3x^3-2x^2-10x+8\)</p>
      </td>
      <td>
        <p>\(f(-3)=-61\)</p>
      </td>
    </tr>
    <tr>
      <td>
        <p>\(x^4-16x^2+3x+15\)</p>
      </td>
      <td>
        <p>\(x - (-4)\)</p>
      </td>
      <td>
        <p>\(3\)</p>
      </td>
      <td>
        <p>\(f(x)=x^4-16x^2+3x+15\)</p>
      </td>
      <td>
        <p>\(f(-4)=3\)</p>
      </td>
    </tr>
  </tbody>
</table>
<br>
<p>To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and adding the remainder. In function notation, we could say: To get the dividend \(f(x)\), we multiply the quotient, \(q(x)\), times the divisor, \(x-c\), and add the remainder, \(r\).</p>
<p>\(f(x)=q(x)(x-c)+r\)</p>
<p> If we evaluate this at \(c\), we get:<br>
  \(f({\style{color:red}c})=q({\style{color:red}c})({\style{color:red}c}-c)+r\)<br>
  \(f(c)=q(c)(0)+r\)<br>
  \(f(c)=r\)</p>
<p>This leads us to the Remainder Theorem.</p>
<br>
<div class="os-raise-graybox">
  <p><strong>REMAINDER THEOREM</strong></p>
  <hr>
  <p>If the polynomial function \(f(x)\) is divided by \(x-c\), then the remainder is \(f(c)\).</p>
</div>
<br>
<p class="os-raise-text-bold">Example 2</p>
<p>Use the Remainder Theorem to find the remainder when \(f(x)=x^3+3x+19\) is divided by \(x + 2\).</p>
<p>To use the Remainder Theorem, we must use the divisor in the \(x-c\) form. We can write the divisor \(x+2\) as \(x-(-2)\). So, our \(c\) is \(-2\).</p>
<p>To find the remainder, we evaluate \(f(c)\), which is \(f(-2)\).</p>
<p>\(f(x)=x^3+3x+19\)</p>
<p><strong>Step 1 -</strong> To evaluate \(f(-2)\), substitute \(x=-2\).</p>
<p> \(f({\style{color:red}-}{\style{color:red}2})=({\style{color:red}-}{\style{color:red}2})^3+3({\style{color:red}-}{\style{color:red}2})+19\)</p>
<p><strong>Step 2 -</strong> Simplify. </p>
<p> \(f(-2)=(-8)-6+19\) <br>
  \(f(-2)=5\)<br>
  The remainder is 5 when \(f(x)=x^3+3x+19\) is divided by \(x+2\).</p>
<p><strong>Step 3 -</strong> Check using synthetic division.</p>
<p><img alt="." src="https://k12.openstax.org/contents/raise/resources/05a284d4b51db6421214ef8ac2c1d86317bb15a1"><br>
  The remainder is 5. </p>
<p>&nbsp;</p>
<h4>Try It: Using the Remainder Theorem</h4>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
  <div class="os-raise-ib-cta-content">
    <p>Use the Remainder Theorem to find the remainder when \(f(x)=x^3+4x+15\) is divided by \(x + 2\).</p>
  </div>
  <div class="os-raise-ib-cta-prompt">
    <p>Write down your answer. Then select the <strong>solution</strong> button to compare your work. </p>
  </div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal2">
  <p>Here is how to solve a problem using the Remainder Theorem:</p>
  <p>\(f(x)=x^3+4x+15\)</p>
  <p>\(f(2)=(-2)^3+4(-2)+15\)</p>
  <p>\(f(2)=-8-8+15= -1\)</p>
  <p>The remainder is \(-1\).</p>
</div>