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<h4>Factor Trinomials</h4>
<p><strong>Example 1</strong></p>
<p>To factor the trinomial \(x^2+8x+12\), you can reverse the process of FOIL.</p>
<p>The trinomial factors to two binomials of the form \((x+m)(x+n)\).</p>
<ul>
<li> Using FOIL: \((x+m)(x+n)=x^2+nx+mx+mn\). </li>
</ul>
<ul>
<li> Using the distributive property: \(x^2+(n+m)x+mn\). </li>
</ul>
<p>Take a look at the trinomial \(x^2+8x+12\):</p>
<p><img alt="IMAGE SHOWING x^2 + (n + m)x + mn and x^2 + 8x + 12 and arrow pointing from n + m to 8, and an arrow from mn to 12." class="img-fluid atto_image_button_text-bottom" height="112" src="https://k12.openstax.org/contents/raise/resources/94e92ec731c8b8a7659a02270f41163aff2fe48e" width="225"></p>
<p>To factor the trinomial, we need factors of 12, \(m\) and \(n\), so that the sum, \(n+m\), is 8.</p>
<p>Use a table to find all the different combinations of factors of 12. Then, find the sum of the factors.</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">Factors of 12</th>
<th scope="col">Sum of factors</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(1, 12\)</p>
</td>
<td>
<p>\(1 + 12 = 13\)</p>
</td>
</tr>
<tr>
<td>
<p>\(2, 6\)</p>
</td>
<td>
<p> \(2 + 6 = 8 ✓\)</p>
</td>
</tr>
<tr>
<td>
<p>\(3, 4\)</p>
</td>
<td>
<p>\(3 + 4 = 7\)</p>
</td>
</tr>
</tbody>
</table>
<br>
<p>The numbers 2 and 6 have a product of 12 and a sum of 8. They are the factors we need:</p>
<p>\(x^2+8x+12=(x+6)(x+2)\)</p>
<p><strong>Example 2</strong></p>
<p>Factor the trinomial \(x^2-5x-24\).</p>
<p>To factor the trinomial, we need factors of -24, \(m\) and \(n\), so that the sum, \(n+m\) is -5.</p>
<ul>
<li> The product of the factors is negative, so the factors must have opposite signs. </li>
<li> The sum is negative, so the negative sign will go with the factor with the greater absolute value. </li>
</ul>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">Factors of 12</th>
<th scope="col">Sum of factors</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(1, −24\)</p>
<p>\(2, −12\)</p>
<p>\(3, −8\)</p>
<p>\(4, −6\)</p>
</td>
<td>
<p>\(1 + (−24) = −23\)</p>
<p>\(2 + (−12) = −10\)</p>
<p>\(3 + (−8) = −5 ✓\)</p>
<p>\(4 + (−6) = −2\)</p>
</td>
</tr>
</tbody>
</table>
<br>
<p>The numbers 3 and -8 have a product of -24 and a sum of -5.</p>
<p>\(x^2-5x-24=(x+3)(x-8)\)</p>
<p><strong>Example 3</strong></p>
<p>Factor the trinomial \(x^2-9x+18\).</p>
<p>To factor the trinomial, we need factors of 18, \(m\) and \(n\), so that the sum, \(n+m\), is -9.</p>
<ul>
<li> The product of the factors is positive, so the factors must have the same signs. </li>
<li> The sum is negative, so both factors are negative. </li>
</ul>
<br>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">Factors of 18</th>
<th scope="col">Sum of factors</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p> \(−1, −18\)</p>
<p> \(−2, −9\)</p>
<p> \(−3, −6\)</p>
</td>
<td>
<p> \(−1 + (−18) = −19\)</p>
<p> \(−2 + (−9) = −11\)</p>
<p> \(−3 + (−6) = −9 ✓\)</p>
</td>
</tr>
</tbody>
</table>
<br>
<p>The numbers -3 and -6 have a product of 18 and a sum of -9.</p>
<p>\(x^2-9x+18=(x-3)(x-6)\)</p>
<h4>Try It: Factoring Trinomials</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">
<p>Factor the trinomial \(x^2+2x-24\).</p>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer. 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><span>Here is how to factor the trinomial \(x^2+2x-24\).</span><br></p>
<p>To factor the trinomial, we need factors of -24, \(m\) and \(n\), so that the sum, \(n+m\), is 2.</p>
<ul>
<li> The product of the factors is negative, so the factors must have opposite signs. </li>
<li> The sum is positive, so the greater factor is positive. </li>
</ul>
<br>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">Factors of 24</th>
<th scope="col">Sum of factors</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\(−1, 24\)</p>
<p>\(−2, 12\)</p>
<p>\(−3, 8\)</p>
<p>\(−4, 6\)</p>
</td>
<td>
<p>\(−1 + 24 = 23\)</p>
<p>\(−2 + 12 = 10\)</p>
<p>\(−3 + 8 = 5\)</p>
<p>\(−4 + 6 = 2 ✓\)</p>
</td>
</tr>
</tbody>
</table>
<br>
<p>The numbers 6 and -4 have a product of -24 and a sum of 2.</p>
<p>\(x^2+2x-24=(x+6)(x-4)\)</p>
</div>
<br>
<h4>Check Your Understanding</h4>
<br>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="d6044b3d-3b2f-4be0-98ef-a98d2159d5eb" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q#-->
<div class="os-raise-ib-pset-problem" data-content-id="8e67e6b6-8ffe-437f-8758-f0876c913476" data-problem-type="multiplechoice" data-solution="\((y-4)(y-9)\)" data-solution-options='["\\((y-4)(y-9)\\)", "\\((y-2)(y+18)\\)", "\\((y+4)(y+9)\\)", "\\((y-6)(y-6)\\)"]'>
<div class="os-raise-ib-pset-problem-content">
<p>Factor: \(y^2-13y+36\).</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct!</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>Take a moment to think about what you learned in the mini-lesson review. </p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is \((y-4)(y-9)\). </p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
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</div>
</div>
<h4>Videos</h4>
<p>Watch the following video to learn more about factoring trinomials.</p>
<p>Khan Academy: Factoring Quadratics</p>
<br>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen frameborder="0" src="https://www.youtube-nocookie.com/embed/1kfq0aR3ASs;&rel=0" title="Factoring quadratics as (x+a)(x+b) (example 2) | Mathematics II | High School Math | Khan Academy"></iframe></div>
</div>