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30b8d5cb-5baa-46b3-aa7c-c41eae8c4948.html
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<h4>Activity </h4>
<p>Take a moment to look at all parts of the diagram. Are there any parts that you do not understand?</p>
<p>Use this time to ask your teacher if there is something you would like clarified here.</p>
<p><img alt height="350" role="presentation"
src="https://k12.openstax.org/contents/raise/resources/279a48d8339d0f5a13e7ba2f062edffc03ef611d"></p>
<p>Use the diagram above as needed for the activity.</p>
<p>Determine the best strategy to factor the polynomial. Be prepared to show your reasoning. Remember to factor out the
GCF first, if necessary, before choosing a strategy. If a polynomial is prime, write “prime.” An example
is provided.</p>
<table class="os-raise-wideequaltable">
<thead>
<tr>
<th scope="col">
Polynomial
</th>
<th scope="col">
Reasoning
</th>
<th scope="col">
Strategy
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(4x^2 + 20xy + 25y^2\)
</td>
<td>
After GCF, trinomial of form \(a^2 + 2ab + b^2\)
</td>
<td>
trinomial square pattern
</td>
</tr>
</tbody>
</table>
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="d46f3813-a21a-4816-88a0-3ddde888d397"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li> \(8y^3 + 16y^2 − 24y\)</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>After GCF, trinomial of form \(x^2+bx+c\); undo FOIL</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="80e2155f-a94c-4dca-bf60-4eef3a0acf24"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="2">
<li> \(16x^3 − 36x\) </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>After GCF, a binomial of squares; difference of squares</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="48689482-47c8-43f2-a6c5-ee15af3bb1c5"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>\(6pq^2 − 9pq − 6p\) </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>After GCF, trinomial of form \(ax^2+bx+c\); “\(ac\)” method or trial and error</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="97bef4ae-4a22-4faf-ad41-34d75026e421"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="4">
<li> \(4x^2 − 12xy + 9y^2 − 25\)</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>polynomial of 4 terms; grouping</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="c70f22ff-fe16-4ad2-80a2-4e004ac34c80"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="5">
<li>\(a^2 + 9\)</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 factors; prime</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="bb23a7f5-1724-46fd-b22f-4e1d4ac2af3f"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="6">
<li> \(9x^2 − 24xy + 16y^2\)</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>trinomial of form \(a^2 − 2ab + b^2\); trinomial square pattern</p>
</div>
</div>
<!--Interaction End -->
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="c7e5b2c0-d3de-4461-b2b4-ed46be6afd77"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="7">
<li> \(27y^2 − 48\)</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>After GCF, a binomial of squares; difference of squares</p>
</div>
</div>
<!--Interaction End -->
<br>
<!-- "Start "Are you Ready for More? click to reveal. If you have more than one on a page, you will need to change the Data-fire/data-wait-for-events for each set.-->
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="RFM1"
data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Start interaction. If multiple interactions appear under "are you ready for more?, they should all have matching "data-wait-for-event", which should also match the "data-fire-event" for the button. Note in this sample, they are all RFM1-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="ce487c66-d011-4d68-b080-4c39d00cdb36"
data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<p>Factor the polynomial below. Write down every factoring strategy that you use.</p>
<p>\(9x^2 − 12xy + 4y^2 − 49\)</p>
</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>\((3x − 2y −7)(3x − 2y + 7)\); Strategies Used: </p>
<p>Factor out GCF, grouping, perfect square trinomial, difference of squares</p>
<p>Here is how tto factor this polynomial:</p>
<p><strong>Step 1 - </strong>Is there a GCF? </p>
<p> \(9x^2−12xy+4y^2−49\)<br>
No.</p>
<p><strong>Step 2 -</strong> With more than three terms, use grouping. The last two terms have no GCF. Try grouping the first three terms.<br>
</p>
<p><strong>Step 3 - </strong>Factor the trinomial with \(a \neq 1\). But the first term is a perfect square.<br>
Is the last term of the trinomial a perfect square? </p>
<p> \((3x)^2−12xy+(2y)^2−49\)</p>
<p> Yes.</p>
<p><strong>Step 4 -
</strong>Does the trinomial fit the pattern \(a^2 \pm 2ab + b^2\)? </p>
<p>
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">
<semantics>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
<msub>
<mrow></mrow>
<mtext>↘</mtext>
</msub>
<munder>
<mrow>
<mn>−12</mn>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
</mrow>
<mrow>
<mrow>
<mn>−2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
</munder>
<msub>
<mrow></mrow>
<mtext>↙</mtext>
</msub>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>49</mn>
</mrow>
</mrow>
<annotation-xml encoding="MathML-Content">
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
<msub>
<mrow></mrow>
<mtext>↘</mtext>
</msub>
<munder>
<mrow>
<mn>−12</mn>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
</mrow>
<mrow>
<mrow>
<mn>−2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
</munder>
<msub>
<mrow></mrow>
<mtext>↙</mtext>
</msub>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>49</mn>
</mrow>
</annotation-xml>
</semantics>
</math>
</p>
<p> Yes. </p>
<p><strong>Step 5 - </strong>Write the trinomial as a square.</p>
<p> \((3x−2y)^2−49\)</p>
<p><strong>Step 6 -</strong> Is this binomial a sum or difference? Of squares or cubes? Write it as a difference of squares.</p>
<p> \((3x−2y)^2−7^2\)</p>
<p><strong>Step 7 -</strong> Write it as a product of conjugates.</p>
<p> \(((3x−2y)−7)((3x−2y)+7)\)<br>
\((3x−2y −7)(3x−2y +7)\)</p>
<p><strong>Step 8 - </strong>Is the expression factored completely? </p>
<p> Yes.</p>
<p><strong>Step 9 -</strong> Check your answer by multiplying. </p>
<p> \((3x−2y−7)(3x−2y+7)\)<br>
\(9x^2−6xy−21x−6xy+4y^2+14y+21x−14y−49\)<br>
\(9x^2−12xy+4y^2−49✓\)</p>
</div>
</div>
<!--End interaction-->