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<h3>Dividing Polynomials Using Synthetic Division<br>
</h3>
<p>Mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in a previous example and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" height="157" role="presentation" src="https://k12.openstax.org/contents/raise/resources/be4336f7e769443f6c877198657487f092f74506" width="550"></p>
<p>Synthetic division basically just removes unnecessary repeated variables and numbers. Here, all the \(x\) and \(x^2\) are removed as well as the \(-x^2\) and \(-4x\) because they are opposite the term above.</p>
<p>The first row of the synthetic division is the coefficients of the dividend. The \(-5\) is the opposite of the \(5\) in the divisor.</p>
<p>The second row of the synthetic division is the numbers shown in red in the division problem.</p>
<p>The third row of the synthetic division is the numbers shown in blue in the division problem.</p>
<p>Notice the quotient and remainder are shown in the third row.</p>
<p><em>Synthetic division only works when the divisor is of the form </em>\(x-c\).</p>
<p>The following example will explain the process.</p>
<p><strong>Example 1</strong></p>
<p>Use synthetic division to find the quotient and remainder when \(2x^3+3x^2+x+8\) is divided by \(x + 2\).</p>
<p><strong>Step 1 -</strong> Write the dividend with decreasing powers of \(x\).</p>
<p> \(2x^3+3x^2+x+8\)</p>
<p><strong>Step 2 -</strong> Write the coefficients of the terms as the first row of the synthetic division.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/605dedc614aaf7172313a858e486e93577dd9c08" alt="." width="298" height="37"> </p>
<p><strong>Step 3 -</strong> Write the divisor as \(x-c\) and place \(c\) in the synthetic division in the divisor box.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/43ddee822d9c8040634e97e271f57ed4ab8ee6a5" alt="." width="298" height="38"> </p>
<p><strong>Step 4 -</strong> Bring down the first coefficient to the third row.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/db7df7f0b07869cce63ee5051e38624b984f3fca" alt="." width="298" height="107"> </p>
<p><strong>Step 5 - </strong>Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/270df054b448712f05e0ece2d4bf879b5c171dca" alt="." width="298" height="107"> </p>
<p><strong>Step 6 -</strong> Add the second column, putting the result in the third row.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/415f13c504557223a7b69f716ef6981993877f93" alt="." width="298" height="108"> </p>
<p><strong>Step 7 - </strong>Multiply that result by the divisor and place the result in the second row under the third coefficient.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/9c08233c731812a9412174d2b36b9a483db062cd" alt="." width="298" height="107"> </p>
<p><strong>Step 8 -</strong> Add the third column, putting the result in the third row.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/f8c6fbb3d3415423a4c9b853cbf5aa14b4fbb0f6" alt="." width="298" height="107"> </p>
<p><strong>Step 9 -</strong> Multiply that result by the divisor and place the result in the third row under the third coefficient.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/bee495ecf46a89d1400bd163c4a8d9963502b4e3" alt="." width="298" height="107"> </p>
<p><strong>Step 10 -</strong> Add the final column, putting the result in the third row.</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/4149128ea5810bc96f9c1c027e17d53e2f4cb927" alt="." width="298" height="108"> </p>
<p><strong>Step 11 - </strong>The quotient is \(2x^2-1x+3\), and the remainder is \(2\).</p>
<p>The division is complete. The numbers in the third row give us the result. The \(2\) \(-1\) \(3\) are the coefficients of the quotient. The quotient is \(2x^2-1x+3\). The \(2\) in the box in the third row is the remainder.</p>
<p>Compare your answer:</p>
<p>
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<p><strong>Example 2</strong></p>
<p>Use synthetic division to find the quotient and remainder when \(x^4-16x^2+3x+12\) is divided by \(x + 4\).</p>
<p>The polynomial \(x^4-16x^2+3x+12\) has its terms in order with descending degree, but we notice there is no \(x^3\) term. We will add a \(0\) as a placeholder for the \(x^3\) term. In \(x - c\) form, the divisor is \(x-(-4)\).</p>
<p><img alt="The figure shows the results of using synthetic division with the example of the polynomial x to the fourth power minus 16 x squared plus 3 x plus 12 divided by x plus 4. The divisor number if negative 4. The first row is 1 0 negative 16 3 12. The first column is 1 blank 1. The second column is negative 16 16 0. The third column is 3 0 3. The fourth column is 12 negative 12 0." src="https://k12.openstax.org/contents/raise/resources/6c98957ba517e1c8bb04a8bba47a6aec12c8cfe3"></p>
<p>We divided a 4th degree polynomial by a 1st degree polynomial, so the quotient will be a 3rd degree polynomial. Reading from the third row, the quotient has the coefficients \(1\) \(-4\) \(0\) \(3\), which is \(x^3-4x^2+3\). The remainder is \(0\).</p>
<h4>Try It: Dividing Polynomials Using Synthetic Division</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">
<ol class="os-raise-noindent">
<li> Use synthetic division to find the quotient and remainder when \(3x^3+10x^2+6x-2\) is divided by \(x + 2\). </li>
</ol>
<ol class="os-raise-noindent" start="2">
<li> Use synthetic division to find the quotient and remainder when \(x^4-16x^2+5x+20\) is divided by \(x + 4\). </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 find these quotients using synthetic division:</p>
<ol class="os-raise-noindent">
<li>\(
3x^2+4x-2\); remainder of \(2\) </li>
</ol>
<p>\(\begin{array}{l}\left. {\underline {\,<br>
{ - 2} \,}}\! \right| \,\,\,\,\,3\,\,\,\,\,\,10\,\,\,\,\,\,6\,\,\,\,\, - 2\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 6\,\,\,\, - 8\,\,\,\,\,\,\,4\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,3\,\,\,\,\,\,4\,\,\,\,\, - 2\,\,\,\,\,\,\left| \!{\overline {\,<br>
2 \,}} \right. \end{array}\)</p>
<ol class="os-raise-noindent" start="2">
<li> \(x^3-4x^2+5\); remainder of \(0\) </li>
</ol>
<p>\(\begin{array}{l}\left. {\underline {\,<br>
{ - 4} \,}}\! \right| \,\,\,\,\,1\,\,\,\,\,\,\,\,0\,\,\,\,\,\, - 16\,\,\,\,\,\,\,5\,\,\,\,\,\,\,\,20\\\underline {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 4\,\,\,\,\,\,\,\,\,16\,\,\,\,\,\,\,0\,\,\,\, - 20\,} \\\,\,\,\,\,\,\,\,\,\,\,\,1\,\,\,\,\, - 4\,\,\,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,5\,\,\,\,\,\,\,\,\,\left| \!{\overline {\,<br>
0 \,}} \right. \end{array}\)</p>
</div>