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<h4>Activity (20 minutes)</h4>
<p>This activity introduces a new method for dividing polynomials. Synthetic division is a process to divide polynomials that streamlines the steps as compared to long division. It works for dividing by any binomial of the form \(x - c\), where \(c\) is a constant. The dividend can be a polynomial with terms of higher degrees.</p>
<p>This lesson represents an extension from the TEKS listed and can be used as an enrichment to enhance the students’ understanding of dividing polynomials.</p>
<h4> Launch</h4>
<p>Ask students to divide the first example, \((x^2-13x+42) \div (x-6)\), using long division. Use the reveal to allow them to check their answer when completed.</p>
<p>Tell students there is a more efficient way to divide a polynomial by a binomial. Emphasize that this method only applies to division of a polynomial by a binomial. Also stress that the divisor must be written in the format \(x - c\), where \(c\) is a constant.</p>
<p>Click the reveal to show the synthetic division process on the same division problem. Guide students through each step, stopping for any clarification as needed.</p>
<p>Arrange the students into pairs. Allow students the remaining time to work through division problems 1–4 in pairs.</p>
<h4>Student Activity </h4>
<p>In this activity, we’re going to use synthetic division to divide a polynomial by a binomial.</p>
<p class="os-raise-text-bold">Example 1</p>
<p>You are going to solve a division problem using long division.</p>
<p> \((x^2-13x+42) \div (x-6)\)</p>
<p>Write down your answer then select the <strong>solution</strong> button to compare your work.</p>
<p class="os-raise-text-bold">Answer: </p>
<p>\[\begin{array}{l}x - 6\mathop{\left){\vphantom{1{{x^2} - 13x + 42}}}\right.<br>
\!\!\!\!\overline{\,\,\,\,\,\,\,\,\,\,\,\,\vphantom 1{{{x^2} - 13x + 42}}}}<br>
\limits^{\displaystyle {x\,\, - \,\,7}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - ({x^2} - 6x)} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 7x + 42\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - ( - 7x + 42)} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\end{array}\]</p>
<p class="os-raise-text-bold">Example 2</p>
<p>You are now going to solve the same division problem using synthetic division.</p>
<p> \((x^2-13x+42) \div (x-6)\)</p>
<p>Select the <strong>Solution</strong> button to show the method.</p>
<p class="os-raise-text-bold">Answer: </p>
<p>\((x^2-13x+42) \div (x-6)\)</p>
<p><strong>Step 1 -
</strong>Write the dividend with decreasing powers of \(x\). Make sure there is no degree missing from the highest degree down to a degree of \(0\).</p>
<p> \(x^2-13x+42\)</p>
<p><strong>Step 2 -
</strong>Write the coefficients of the terms as the first row of the synthetic division.</p>
<p><img height="25" src="https://k12.openstax.org/contents/raise/resources/c6c82cbc2862fbc7ac63ef5ad156a357e28d2929" width="200">
</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 height="26" src="https://k12.openstax.org/contents/raise/resources/f842fbc8fba49798644a0db226343949a76160d2" width="200">
</p>
<p><strong>Step 4 -</strong> Bring down the first coefficient to the third row.</p>
<p><img height="78" src="https://k12.openstax.org/contents/raise/resources/360673e02e9a7c448ea49dbe61338082bf855e69" width="200">
</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 height="79" src="https://k12.openstax.org/contents/raise/resources/61bd9ababefd8697792d305cad39a9ef67d99e8e" width="200">
</p>
<p><strong>Step 6 -
</strong>Add the second column, putting the result in the third row.</p>
<br>
<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 alt class="img-fluid atto_image_button_text-bottom" role="presentation" src="https://k12.openstax.org/contents/raise/resources/32c4276bc6a13be7a299d2772d58550666421700" width="200"></p>
<p><strong>Step 8 -</strong> Add the final column, putting the result in the third row.</p>
<p><img alt class="img-fluid atto_image_button_text-bottom" role="presentation" src="https://k12.openstax.org/contents/raise/resources/6616be07a9b6a0afd9f2bda3795506fe71dbfb29" width="200">
</p>
<p><strong>Step 9 -</strong> The division is complete. The numbers in the third row give us the result. The \(1\) \( −7\) in the third row are the coefficients of the quotient. The quotient is \(x - 7\). The \(0\) in the box in the third row is the remainder.</p>
<br>
<div class="os-raise-graybox">
<p>Have students, work with a partner to find each quotient. Discuss any differences you have in determining the correct solution.</p>
</div>
<br>
<ol class="os-raise-noindent" start="3">
<li>\(
(4x^3+5x^2-5x+3) \div (x+2)\) </li>
</ol>
<p><strong>Answer:</strong> \(4x^2-3x+1\); remainder of 1</p>
<ol class="os-raise-noindent" start="4">
<li> \((2x^3-11x^2+16x-12) \div (x -4)\) </li>
</ol>
<p><strong>Answer: </strong>\(2x^2-3x+4\); remainder of 4</p>
<ol class="os-raise-noindent" start="5">
<li>\(
(x^4-9x^2+2x+6) \div (x + 3)\) </li>
</ol>
<p><strong>Answer: </strong>\(x^3-3x^2+2\); remainder of 0</p>
<ol class="os-raise-noindent" start="6">
<li> \((3x^4-11x^3+2x^2+10x+6) \div (x - 3)\) </li>
</ol>
<p><strong>Answer: </strong>\(3x^3-2x^2-4x-2\); remainder of 0</p>
<h4> Anticipated Misconceptions</h4>
<p>It is important for students to recognize that synthetic division only works when the divisor is of the form \(x - c\). If the divisor is already in the form \(x - c\), such as \(x - 3\), then \(3\) will go into the box in the top-left corner of the synthetic division. If the divisor is \(x + 5\), then it must be converted into the form \(x - c\) by adding the appropriate signs to maintain equivalency. Since \(x + 5 = x - (-5)\), \(-5\) would go into the box in the top-left corner.</p>
<h4>Activity Synthesis</h4>
<p>Remember these key points to help students summarize the lesson:</p>
<ul>
<li> Synthetic division is only useful when the divisor is a binomial of the form \(x - c\), where \(c\) is a constant. </li>
<li> The dividend can be a polynomial with terms of higher degrees. When completing synthetic division, arrange the terms of the dividend in descending order according to degree. Add placeholders with a \(0\) coefficient for any terms that are missing, such as \(0x^3\). </li>
<li> The final number in the third row is the remainder. The remainder is expressed as a fraction with the value from the synthetic division as the numerator and the divisor as the denominator. </li>
</ul>
<h3> 6.3.3: Self Check</h3>
<!--BEGIN SELF CHECK INTRO BEFORE Tables -->
<p class="os-raise-text-bold os-raise-text-italicize">After the activity, students will answer the following question to check their understanding of the concepts explored in the activity.</p>
<!--SELF CHECK QUESTION GOES BEFORE THE Table -->
<p class="os-raise-text-bold">QUESTION:</p>
<p>Use synthetic division to find the quotient and remainder when \(x^4+x^2+6x-10\) is divided by \(x + 2\).</p>
<!--SELF CHECK table-->
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(x^3-x^2+8x\); remainder of \(26\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: Remember to use \(0x^3\) as a placeholder in the dividend. The answer is \(x^3-2x^2+5x-4\); remainder of \(-2\).</p>
</td>
</tr>
<tr>
<td>
<p>\(x^3-2x^2+5x-4\); remainder of \(0\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: Check your calculation on the remainder. The answer is \(x^3-2x^2+5x-4\); remainder of \(-2\).</p>
</td>
</tr>
<tr>
<td>
<p>\(x^3-2x^2+5x-4\); remainder of \(2\)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: Check your calculation on the remainder. The answer is \(x^3-2x^2+5x-4\); remainder of \(-2\).</p>
</td>
</tr>
<tr>
<td>
<p>\(x^3-2x^2+5x-4\); remainder of \(-2\)</p>
</td>
<td>
<p>That’s correct! Check yourself: Multiply \((x^3-2x^2+5x-4)(x+2)\) and add \(-2\). Since this equals \(x^4+x^2+6x-10\), it is correct.</p>
</td>
</tr>
</tbody>
</table>
<br>
<!--END SELF CHECK INTRO BEFORE Tables -->
<br>
<h3>6.3.3: Additional Resources</h3>
<p class="os-raise-text-bold os-raise-text-italicize">The following content is available to students who would like more support based on their experience with the self check. Students will not automatically have access to this content, so you may wish to share it with those who could benefit from it.</p>
<h4>Dividing Polynomials Using Synthetic Division</h4>
<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>
<br>
<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>
<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 src="https://k12.openstax.org/contents/raise/resources/6c98957ba517e1c8bb04a8bba47a6aec12c8cfe3" 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." width="398" height="107"></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>
<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>
<p>Write down your answers. Then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<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>