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8d7f7571-c3f0-4ca7-a1ba-31c523d636da.html
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<h4>Activity </h4>
<p>Let’s learn a new method that always works on any trinomial of the form \(ax^2 + bx + c\).</p>
<br>
<div class="os-raise-graybox">
<p> <strong> How to factor trinomials of the form \(ax^2 + bx + c\) using the “\(ac\)” method: </strong></p>
<hr>
<p><strong>Step 1 -</strong> Factor any GCF. Don’t forget!</p>
<p><strong>Step 2 -</strong> Find the product \(ac\).</p>
<p><strong>Step 3 - </strong>Find two numbers, \(m\) and \(n\), that</p>
<ul>
<li>multiply to \(ac\): \(m \cdot n = a \cdot c\).</li>
<li>add to \(b\): \(m + n = b\).</li>
</ul>
<p><strong>Step 4 -</strong> Split the middle term using \(m\) and \(n\).</p>
<p> \(ax^2 + mx + nx + c\)</p>
<p><strong>Step 5 -</strong> Factor by grouping.</p>
<p><strong>Step 6 -</strong> Check by multiplying the factors.</p>
</div>
<br>
<p>Factor each trinomial using the “\(ac\)” method.</p>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="4ccfa6de-640b-4f95-b92b-2e02d7defa41" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li> \(16x^2 − 32x + 12\) </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>\(4(2x − 3)(2x − 1)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="85f50172-cb66-4ec0-a8e6-dcd8261a339c" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li> \(6y^2 + y − 15\)</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>\((3y + 5)(2y − 3)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="731967ec-5156-4538-8049-8ee435fbae97" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li> \(48z^3 − 102z^2 − 45z\)</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>\(3z(8z + 3)(2z − 5)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<p>There will be times when a trinomial does not seem to be in the form \(ax^2 + bx + c\).</p>
<p>In some of these cases, we can make a substitution that allows the trinomial to be written in the \(ax^2 + bx + c\) form. This is called factoring by substitution.</p>
<p>It is standard to use \(u\) for the substitution. The substitution for \(u\) might be a variable with an exponent, such as \(x^2\), or it might be a binomial, such as \((x − 3)\).</p>
<p>Factor each trinomial using the substitution method.</p>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="79d498de-a076-425c-870d-4eb29109ab29" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li> \(y^4 − y^2 − 20\)</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>\((y^2 + 4)(y^2 − 5)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="62847554-78c6-4dff-a335-310ae419febc" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="5">
<li>\((x−5)^2 + 6(x − 5) + 8\)</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>\((x − 3)(x − 1)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="a4073d91-2353-48af-952b-177e95e8fbfd" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li> \(x^4 − 3x^2 − 28\)</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>\((x^2 − 7)(x^2 + 4)\) </p>
</div>
</div>
<!--Interaction End -->