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<h4>Implementing General Strategies for Factoring</h4>
<p>Let’s practice recognizing and implementing these strategies as we factor polynomials.</p>
<p>Remember, a polynomial is completely factored if, other than monomials, its factors are prime!</p>
<p class="os-raise-text-bold">EXAMPLE 1</p>
<p>Find a strategy that fits and factor completely: \(7x^3 − 21x^2 − 70x\).</p>
<p><strong>Step 1</strong> - Is there a GCF?<br>
\(7x^3 − 21x^2 − 70x\)<br>
Yes, \(7x\).</p>
<p><strong>Step 2</strong> - Factor out the GCF.<br>
\(7x(x^2 − 3x − 10)\)</p>
<p><strong>Step 3</strong> - In the parentheses, is it a binomial, a trinomial, or are there more than three terms?<br>
Trinomial with leading coefficient \(1\).</p>
<p><strong>Step 4</strong> - “Undo” FOIL.<br>
\(7x(x \;\;\;)(x \;\;\; )\)<br>
\(7x(x + 2)(x − 5)\)</p>
<p><strong>Step 5</strong> - Is the expression factored completely? <br>
Yes. Neither binomial can be factored.</p>
<p><strong>Step 6</strong> - Check your answer by multiplying.<br>
\(7x(x + 2)(x − 5)\)<br>
\(7x(x^2 − 5x + 2x − 10)\)<br>
\(7x(x^2 − 3x − 10)\)<br>
\(7x^3 − 21x^2 − 70x✓\)</p>
<p class="os-raise-text-bold">EXAMPLE 2</p>
<p>Find a strategy that fits and factor completely: \(4x^2 + 8bx − 4ax − 8ab\).</p>
<p><strong>Step 1</strong> - Is there a GCF?<br>
\(4x^2 + 8bx − 4ax − 8ab\)<br>
Factor out the GCF, \(4\).<br>
\(4(x^2 + 2bx − ax − 2ab)\)</p>
<p><strong>Step 2</strong> - There are four terms. Use grouping.<br>
\(4 \lbrack x(x + 2b) − a(x + 2b) \rbrack\)<br>
\(4(x + 2b)(x − a)\)</p>
<p><strong>Step 3</strong> - Is the expression factored completely?<br>
Yes.</p>
<p><strong>Step 4</strong> - Check your answer by multiplying. <br>
\(4(x + 2b)(x − a)\) <br>
\(4(x^2 − ax + 2bx − 2ab)\) <br>
\(4x^2 + 8bx − 4ax − 8ab✓\) </p>
<p class="os-raise-text-bold">EXAMPLE 3</p>
<p>Find a strategy that fits and factor completely: \(40x^2y + 44xy − 24y\).</p>
<p><strong>Step 1</strong> - Is there a GCF? <br>
\(40x^2y + 44xy − 24y\) <br>
Factor out the GCF, \(4y\). <br>
\(4y(10x^2 + 11x − 6)\) </p>
<p><strong>Step 2</strong> - Factor the trinomial with \(a \neq 1\) using the “\(ac\)” method or trial and error.<br>
\(4y(5x − 2)(2x + 3)\)</p>
<p><strong>Step 3</strong> - Is the expression factored completely? <br>
Yes. </p>
<p><strong>Step 4</strong> - Check your answer by multiplying. <br>
\(4y(5x − 2)(2x + 3)\) <br>
\(4y(10x^2 + 11x − 6)\) <br>
\(40x^2y + 44xy − 24y✓\) </p>
<h4>Try It: Implementing General Strategies for Factoring</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>Find a strategy that fits and factor completely: </p>
<ol class="os-raise-noindent">
<li>\(27y^2 − 48\) </li>
</ol>
<ol class="os-raise-noindent" start="2">
<li> \(6x^2 − 12xc + 6bx − 12bc\). </li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answers. 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>Here is how to factor these polynomials:</p>
<ol class="os-raise-noindent">
<li> difference of squares; \(3(3y − 4)(3y + 4)\) </li>
</ol>
<ol class="os-raise-noindent" start="2">
<li> grouping; \(6(x + b)(x − 2c)\)</li>
</ol>
</div>