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48c420b9-4af1-4ed9-8ccf-0978edd8b73f.html
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<h4>Activity</h4>
<p>If there is no common term in all of the terms of a polynomial, it may be possible to factor by grouping. To do this, separate the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find that a common factor emerges from both parts.</p>
<p>Let’s look at an example. Factor the following polynomial.</p>
<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>\(ms − 4m + 7s − 28\)</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>Compare your answer:</p>
<p><strong>Step 1 - </strong>Group terms with common factors.</p>
<p> Is there a GCF of all four terms? No, so let’s separate the first two terms from the second two.</p>
<p> \(ms − 4m + 7s − 28\)<br>
<img alt class="img-fluid atto_image_button_text-bottom" height="28" role="presentation" src="https://k12.openstax.org/contents/raise/resources/1b9de1f635412506c067211646613e19a4b113db" width="200"> </p>
<p><strong>Step 2 - </strong>Factor out the common factor in each group.</p>
<p> Factor the GCF from the first two terms. Factor the GCF from the second two terms.</p>
<p> \(m(s − 4) + 7s − 28\)<br>
\(m(s − 4) + 7(s − 4)\)</p>
<p><strong>Step 3 - </strong>Factor the common factor from the expression.</p>
<p> Notice that each term has a common factor of \((s − 4)\). Factor out the common factor.</p>
<p> \(m
{\style{color:red}(}
{\style{color:red}s} \;{\style{color:red}-}\; {\style{color:red}4}
{\style{color:red})}
+ 7{\style{color:red}(}
{\style{color:red}s}\; {\style{color:red}-}\; {\style{color:red}4}
{\style{color:red})}\)<br>
\((s − 4)(m + 7)\)</p>
<p><strong>Step 4 - </strong>Check.</p>
<p> Multiply \((s − 4)(m + 7)\). Is the product the original expression?</p>
<p> \((s − 4)(m + 7)\)<br>
\(ms +7s − 4m − 28\)<br>
\(ms − 4m + 7s − 28\)
\({\style{color:red}✓}\)</p>
</div>
<br>
<p>Work on the following exercises individually.</p>
<p>Factor by grouping:</p>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="b42bc36d-f107-43b0-bc6f-0a25bdf95fde" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li>
<p>\(xy + 7y + 8x + 56\)</p>
</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>\((x + 7)(y + 8)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="793e9b45-b4b0-4ae6-b2d3-ec1006806d9a" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li>
<p>\(x^2 + 2x − 5x − 10\)</p>
</li>
<p>You can use the following video if you need some help</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/b204443db93c94ab3591290f6f435f5498fbd228">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/8b34c3c7ea2318bc967925932e4957127ac286c6" srclang="en_us">
</video></div>
</div>
</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>\((x − 5)(x + 2)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="77909510-5a2f-4919-9a57-b26bc00529ae" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>
<p>\(20x^2 − 16x − 15x + 12\)</p>
</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>\((5x − 4)(4x − 3)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="2c87960b-3434-49f9-a275-1af8c599adc9" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li>
<p>\(x^2 + 4x − 7x − 28\)</p>
</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>\((x + 4)(x − 7)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="cf2fa676-8fe0-4be2-81f8-bd73282b6934" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="5">
<li>
<p>\(2x^2 − 14x − 5x + 35\)</p>
</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>\((x − 7)(2x − 5)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="d46293cd-9bbd-4d55-af1a-1938c38bc7d1" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li>
<p>\(xy − 6x − 4y + 24\)</p>
</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>\((y − 6)(x − 4)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="1cbe17e0-f460-4f84-873a-79b7b629b590" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="7">
<li>
<p>\(9x^2 + 12x − 15x − 20\)</p>
</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>\((3x − 5)(3x + 4)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="1a88296e-b62c-4157-974a-d802f941e995" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="8">
<li>
<p>\(xy − 9x + 2y − 18\)</p>
</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>\((x + 2)(y − 9)\)</p>
</div>
</div>
<!--Interaction End -->
<br>
<h4>Video: Factoring by Grouping</h4>
<p>Watch the following video to learn more about how to factor by grouping.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/93677f51e4bf0e85e72e43e92ec87173fbced0a1">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/cb519836bf0951ce50170e8eadf4589062c8bda6" srclang="en_us">https://k12.openstax.org/contents/raise/resources/93677f51e4bf0e85e72e43e92ec87173fbced0a1
</video></div>
</div>