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<h4>Activity (20 minutes)</h4>
<p>This activity takes the steps students have learned so far, factoring expressions and finding their GCF, and adds the
step of factoring the GCF from the polynomial.</p>
<p>Having a mastery of the Distributive Property is crucial for understanding how it can be “reversed” to
factor the GCF out of each expression in a polynomial.</p>
<h4>Launch</h4>
<p>Start the lesson by reviewing the Distributive Property with students.</p>
<div class="os-raise-graybox">
<p class="os-raise-text-bold"> DISTRIBUTIVE PROPERTY </p>
<hr>
<p>If \(a\), \(b\), and \(c\) are real numbers, then</p>
<p>\(\;\;\;\;\;\;\;\;\;\;\;a(b + c) = ab + ac\) and \(ab + ac = a(b + c)\)</p>
<p>The form on the left is used to multiply. The form on the right is used to factor.</p>
</div>
<br>
<p>Emphasize that the equation on the left is often used to distribute a value into a group of expressions. The equation
on the right can be thought of as the Distributive Property “in reverse” and will be very useful in
factoring expressions.</p>
<p>Use 7–8 minutes to review the example at the start of the activity. As students have already performed many of
the steps for factoring the values and finding the GCF, ask them to explain the first steps as far as they can. Guide
them through any missing steps.</p>
<p>Finally, ask students to complete questions 1–8 individually. If needed, solve the first one together as a
class.</p>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name"><!--Extra Support Name-->MLR 2 Collect and Display: Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<!--Support Content-->
<p>As students describe how to factor and find the GCF with the class, listen for and collect the language students use to identify and describe the factoring process. Write the students' words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions. </p>
<p class="os-raise-text-italicize"><!--Support Italics-->Design Principle(s): Maximize meta-awareness</p>
<p class="os-raise-extrasupport-title"><!--Extra Support Title-->Provide support for students</p>
<!-- Extra Support Content-->
<p><a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a> to the students to assist them with participating in this routine. </p>
</div>
</div>
<br>
<h4>Student Activity </h4>
<p>To factor a GCF from a polynomial and represent it in factored form, we must first find the GCF.</p>
<p>Let’s look at an example: \(7z^5 − 21sz^3\).</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><strong>Step 1 -
</strong>Find the GCF of the terms in the polynomial:</p>
<p><img height="59" src="https://k12.openstax.org/contents/raise/resources/799e3fde5987eae5c5b76e66ec0a7755b163eef9"
width="300"></p>
<p>The GCF of the terms is \(7z^3\).</p>
<p><strong>Step 2 -
</strong>Apply the Distributive Property “in reverse” to factor out the GCF from each term.</p>
<p>\(7z^5 − 21sz^3 = {\style{color:red}7}
{\style{color:red}z}
^
{\style{color:red}3} \cdot z^2 − {\style{color:red}7}
{\style{color:red}z}
^
{\style{color:red}3} \cdot 3s\)</p>
<p>\(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= 7z^3(z^2 − 3s)\)</p>
<p>The factored form is \(7z^3(z^2 − 3s)\).</p>
<p>Work individually on the following problems.</p>
<p>Factor the GCF from each polynomial. Remember that if the leading coefficient is negative, then the GCF will be
negative.</p>
<ol class="os-raise-noindent">
<li> \(5x^3 − 25x^2\) </li>
</ol>
<p><strong>Answer:</strong> \(5x^2(x − 5)\)</p>
<ol class="os-raise-noindent" start="2">
<li> \(8x^3y − 10x^2y^2 + 12xy^3\) </li>
</ol>
<p><strong>Answer:</strong> \(2xy(4x^2 − 5xy + 6y^2)\)</p>
<ol class="os-raise-noindent" start="3">
<li> \(−5y^3 + 35y^2 − 15y\) </li>
</ol>
<p><strong>Answer:</strong> \(−5y(y^2 − 7y + 3)\)</p>
<ol class="os-raise-noindent" start="4">
<li> \(15x^3y − 3x^2y^2 + 6xy^3\) </li>
</ol>
<p><strong>Answer:</strong> \(3xy(5x^2 − xy + 2y^2)\)</p>
<ol class="os-raise-noindent" start="5">
<li> \(−4p^3q − 12p^2q^2 + 16pq^2\) </li>
</ol>
<p><strong>Answer:</strong> \(−4pq(p^2 + 3pq − 4q)\)</p>
<ol class="os-raise-noindent" start="6">
<li> \(5x(x + 1) + 3(x + 1)\) </li>
</ol>
<p><strong>Answer:</strong> \((x + 1)(5x + 3)\)</p>
<ol class="os-raise-noindent" start="7">
<li> \(3b(b − 2) − 13(b − 2)\) </li>
</ol>
<p><strong>Answer:</strong> \((b − 2)(3b − 13)\)</p>
<ol class="os-raise-noindent" start="8">
<li> \(6m(m − 5) − 7(m − 5)\) </li>
</ol>
<p><strong>Answer:</strong> \((m − 5)(6m − 7)\)</p>
<h4>Anticipated Misconceptions</h4>
<p>Students may run into trouble in the activity if they are unable to find the correct GCF of the expressions in the
polynomial. Students who do not list each prime factor for each expression may not completely factor those
expressions. This will lead to finding a common factor that is not the greatest common factor. After factoring out
their common factor, the polynomial will not be completely factored. Emphasize that listing the prime factors for each
expression is a very important step in the process.</p>
<h4>Activity Synthesis</h4>
<p>Ask a student to describe each step in factoring a GCF from a polynomial. Use the list below to help the student as
needed:</p>
<ul>
<li><strong>STEP 1 - </strong>Find the GCF of all the terms in the polynomial.</li>
<li><strong>STEP 2 -</strong> Rewrite each term as a product using the GCF.</li>
<li><strong>STEP 3 - </strong> Use the “reverse” of the Distributive Property to factor the expression.</li>
<li><strong>STEP 4 - </strong> Check your answer by multiplying the factors together.</li>
</ul>
<h3>6.4.3: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their
understanding of the
concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Factor: \(8a^3b + 2a^2b^2 − 6ab^3\).</p>
<table class="os-raise-textheavytable">
<caption>
</caption>
<thead>
<tr>
<th scope="col"> Answers </th>
<th scope="col"> Feedback </th>
</tr>
</thead>
<tbody>
<tr>
<td> \(−2ab(4a^2 + ab − 3b^2)\) </td>
<td> Incorrect. Let’s try again a different way: You should not have factored out a negative sign in the GCF.
The correct answer is \(2ab(4a^2 + ab − 3b^2)\). </td>
</tr>
<tr>
<td> \(ab(8a^2 + 2ab − 6b^2)\) </td>
<td> Incorrect. Let’s try again a different way: The GCF you found is missing the factor 2. The correct answer
is \(2ab(4a^2 + ab − 3b^2)\). </td>
</tr>
<tr>
<td> \(2b(4a^3 + a^2b − 3ab^2)\) </td>
<td> Incorrect. Let’s try again a different way: The GCF you found is missing the factor \(a\). The correct
answer is \(2ab(4a^2 + ab − 3b^2)\). </td>
</tr>
<tr>
<td> \(2ab(4a^2 + ab − 3b^2)\) </td>
<td> That’s correct! Check yourself: Multiply \((2ab)(4a^2 + ab − 3b^2)\) to see if it equals \(8a^3b +
2a^2b^2 − 6ab^3\). Since it does, the answer is correct. </td>
</tr>
</tbody>
</table>
<br>
<h3>6.4.3: 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>Factoring the GCF from Polynomials</h4>
<p>It is sometimes useful to represent a number as a product of factors, for example, 12 as \(2 \cdot 6\) or \(3 \cdot
4\). In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such
as \(3x^2+15x\), and end with its factors, \(3x(x + 5)\). To do this, we apply the Distributive Property “in
reverse.” We state the Distributive Property here just as you saw it in earlier chapters and “in
reverse.”</p>
<div class="os-raise-graybox">
<p class="os-raise-text-bold"> DISTRIBUTIVE PROPERTY </p>
<hr>
<p>If \(a\), \(b\), and \(c\) are real numbers, then</p>
<p>\(\;\;\;\;\;\;\;\;\;\;\;a(b + c) = ab + ac\) and \(ab + ac = a(b + c)\)</p>
<p>The form on the left is used to multiply. The form on the right is used to factor.</p>
</div>
<br>
<p>So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write
the polynomial as a product!</p>
<p class="os-raise-text-bold">Example 1</p>
<p>Factor: \(8m^3-12m^2n+20mn^2\).</p>
<p class="os-raise-text-bold">How to factor the greatest common factor from a polynomial:</p>
<p><strong>Step 1 -
</strong>Find the GCF of all the terms of the polynomial. For this problem, find the GCF of \(8m^3\), \(12m^2n\), and
\(20mn^2\).</p>
<p><img height="135" src="https://k12.openstax.org/contents/raise/resources/2af0ae8a08f2a69dfa71f2cf930dc371ca3a0761"
width="300"></p>
<p><strong>Step 2 -</strong> Rewrite each term as a product using the GCF. In this case, rewrite \(8m^3\), \(12m^2n\), and \(20mn^2\) as products
of their GCF, \(4m\).</p>
<ul>
<li> \(8m^3 = {\style{color:red}4}
{\style{color:red}m} \cdot 2m^2\) </li>
<li> \(12m^2n = {\style{color:red}4}
{\style{color:red}m} \cdot 3mn\) </li>
<li> \(20mn^2 = {\style{color:red}4}
{\style{color:red}m} \cdot 5n^2\) </li>
</ul>
<p>\(8m^3-12m^2n+20mn^2\)<br>
\({\style{color:red}4}
{\style{color:red}m} \cdot 2m^2 - {\style{color:red}4}
{\style{color:red}m} \cdot 3mn + {\style{color:red}4}
{\style{color:red}m} \cdot 5n^2\)</p>
<p><strong>Step 3 -
</strong>Use the “reverse” Distributive Property to factor the expression.</p>
<p>\(4m(2m^2-3mn+5n^2)\)</p>
<p><strong>Step 4 -</strong> Check by multiplying the factors.</p>
<p>\(4m(2m^2-3mn+5n^2)\)<br>
\(4m \cdot 2m^2 - 4m \cdot 3mn + 4m \cdot 5n^2\)<br>
\(8m^3-12m^2n+20mn^2\)</p>
<p>When the leading coefficient is negative, we factor the negative out as part of the GCF.</p>
<p class="os-raise-text-bold">Example 2</p>
<p>Factor: \(−4a^3 + 36a^2 − 8a\).</p>
<p>The leading coefficient is negative, so the GCF will be negative.</p>
<p><strong>Step 1 -</strong> Rewrite each term using the GCF, \(−4a\).</p>
<p> \(
{\style{color:red}-}
{\style{color:red}4}
{\style{color:red}a} \cdot a^2 −
{\style{color:red}(}
{\style{color:red}-}
{\style{color:red}4}
{\style{color:red}a}
{\style{color:red})} \cdot 9a +
{\style{color:red}(}{\style{color:red}-}
{\style{color:red}4}
{\style{color:red}a}
{\style{color:red})} \cdot 2\) </p>
<p><strong>Step 2 -</strong> Factor the GCF.</p>
<p> \(−4a(a^2 − 9a + 2)\)</p>
<p><strong>Step 3 -</strong> Check.</p>
<p> \(−4a(a^2 − 9a + 2)\)<br>
\(−4a \cdot a^2 − (−4a) \cdot 9a + (−4a) \cdot 2\)<br>
\(−4a^3 + 36a^2 − 8a\)</p>
<p>So far, our greatest common factors have been monomials. In the next example, the greatest common factor is a
binomial.</p>
<p class="os-raise-text-bold">Example 3</p>
<p>Factor: \(3y
{\style{color:red}(}
{\style{color:red}y} \;{\style{color:red}+}\; {\style{color:red}7}
{\style{color:red})}
− 4{\style{color:red}(}
{\style{color:red}y}\; {\style{color:red}+} \;{\style{color:red}7}
{\style{color:red})}\).</p>
<p>The GCF is the binomial \(y + 7\).</p>
<p>\(3y(y + 7) − 4(y + 7)\).</p>
<p><strong>Step 1 -</strong> Factor the GCF, \((y + 7)\).</p>
<p> \((y + 7)(3y − 4)\)</p>
<p><strong>Step 2 -</strong> Check on your own by multiplying.</p>
<br>
<h4>Try It: Factoring the GCF from Polynomials</h4>
<p>Factor.</p>
<ol class="os-raise-noindent">
<li> \(9xy^2 + 6x^2y^2 + 21y^3\) </li>
</ol>
<ol class="os-raise-noindent" start="2">
<li> \(−4b^3 + 16b^2 − 8b\) </li>
</ol>
<ol class="os-raise-noindent" start="3">
<li> \(4m(m + 3) − 7(m + 3)\) </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 factor the GCF from polynomials.</p>
<ol class="os-raise-noindent">
<li> \(3y^2(3x + 2x^2 + 7y)\) </li>
</ol>
<ol class="os-raise-noindent" start="2">
<li> \(−4b(b^2 − 4b + 2)\) </li>
</ol>
<ol class="os-raise-noindent" start="3">
<li> \((m + 3)(4m − 7)\) </li>
</ol>