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<h4>Cool Down Activity</h4>
<p>In all the division problems we completed that had a remainder of zero, we say that the divisor was a factor of the dividend.</p>
<p>Whenever a divisor, \(x-c\), divides a polynomial function, \(f(x)\), and results in a remainder of zero, we say \(x-c\) is a factor of \(f(x)\).</p>
<p>This leads us to the Factor Theorem.</p>
<br>
<div class="os-raise-graybox">
<p><strong>FACTOR THEOREM</strong></p>
<hr>
<p>For any polynomial function \(f(x)\),</p>
<ul>
<li> if \(x-c\) is a factor of \(f(x)\), then \(f(c)=0\).</li>
<li> if \(f(c)=0\), then \(x-c\) is a factor of \(f(x)\).</li>
</ul>
</div>
<br>
<p>Use the Factor Theorem to determine whether the binomial is a factor of the given polynomial function. Be prepared to show your reasoning.</p>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="f2d19538-394c-408c-90c6-2fdafcd67d1f" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li> Is \((x - 5)\) a factor of \(f(x)=x^3-125\)?</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Yes, because \(f(5) = (5)^3 - 125 = 125 - 125 = 0\) </p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="356be59a-c6e0-44a7-8648-70883341a215" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li> Is \((x - 6)\) a factor of \(f(x)=x^3-216\)?</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Yes, because \(f(6) = (6)^3 - 216 = 216 - 216 = 0\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="7fe92888-c7c8-4cb9-a80d-5ce56153daef" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li> Is \((x + 3)\) a factor of \(f(x)=x^3+8x^2+21x+18\)?</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Yes, because \(f(-3) = (-3)^3 + 8(-3)^2 + 21(-3) + 18 = -27 + 72 - 63 + 18 = 0\) </p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="b7044937-9059-4ad5-a627-0e7083a185a1" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li> Is \((x + 4)\) a factor of \(f(x)=x^3+x^2-14x+8\)?<br>
</li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>No, because \(f(-4) = (-4)^3 + (-4)^2 - 14(-4) + 8 = -64 + 16 + 56 + 8 = 16\)</p>
</div>
</div>
<!--Interaction End -->