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<h4>Multiplying a Polynomial by a Polynomial</h4>
<p>We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use other representations of the Distributive Property such as Vertical Alignment or distributing individual terms or polynomials.</p>
<br>
<p><strong>Example 1</strong></p>
<p>Multiply \((b+3)(2b^2-5b+8)\) using the Distributive Property by distributing either a binomial or an individual term.</p>
<p><strong>Step 1 - </strong>
Distribute.<br> \({\style{color:red}(}
{\style{color:red}2}
{\style{color:red}b}
^
{\style{color:red}2} \;{\style{color:red}-}\; {\style{color:red}5}
{\style{color:red}b}\; {\style{color:red}+} \;{\style{color:red}8}
{\style{color:red})}\).<br>
\(b
{\style{color:red}(}
{\style{color:red}2}
{\style{color:red}b}
^
{\style{color:red}2}\; {\style{color:red}-}\; {\style{color:red}5}
{\style{color:red}b}\; {\style{color:red}+}\; {\style{color:red}8}
{\style{color:red})}
+3
{\style{color:red}(}
{\style{color:red}2}
{\style{color:red}b}
^
{\style{color:red}2} \;{\style{color:red}-}\; {\style{color:red}5}
{\style{color:red}b} \;{\style{color:red}+}\; {\style{color:red}8}
{\style{color:red})}
\)</p>
<p><strong>Step 2 - </strong>
Multiply.<br>
\(2b^3-5b^2+8b+6b^2-15b+24\)</p>
<p><strong>Step 3 - </strong>
Combine like terms.<br>
\(2b^3+b^2-7b+24\)</p>
<br>
<p><strong>Example 2</strong></p>
<p> Multiply \((b+3)(2b^2-5b+8)\) using vertical alignment. </p>
<p> It is easier to put the polynomial with fewer terms on the bottom because we get fewer partial products this way.</p>
<p><strong>Step 1 - </strong>
Multiply. <br>\((2b^2-5b+8)\) by \(3\).</p>
<p><strong>Step 2 - </strong>
Multiply.<br> \((2b^2-5b+8)\) by \(b\).</p>
<p><strong>Step 3 - </strong>
Add like terms.</p>
<p>\(\underline{\underline{\;\;\;\;\;\;\;\;\;\;2b^2-\;5b+8\\\;\;\;\;\;\;\;\;\;\times\;\;\;\;\;\;\;\;b+3}\\ \; \\\;\;\;\;\;\;\;\;\;6b^2-15b+24\\2b^3-5b^2\;+8b}\\ \; \\ 2b^3 \; +\;b^2-\;7b+24\)</p>
<br>
<h4>Try It: Multiplying a Polynomial by a Polynomial</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>For questions 1 - 2, Multiply \((z-3)(z^2-5z+2)\) using the listed method.</p>
<ol class="os-raise-noindent">
<li> The Distributive Property </li>
</ol>
</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: You answer may vary, but here is a sample.</p>
<p><strong>Step 1 - </strong>Distribute.<br> \((z^2 - 5z + 2)\).<br>
\(z(z^2-5z+2)-3(z^2-5z+2)\)</p>
<p><strong>Step 2 - </strong>
Multiply.<br>
\(z^3-5z^2+2z-3z^2+15z-6\)</p>
<p><strong>Step 3 - </strong>
Combine like terms.<br>
\(z^3-8z^2+17z-6\)</p>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li> Vertical Alignment </li>
</ol>
</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="Reveal2">
<p>Compare your answer: You answer may vary, but here is a sample.</p>
<br>
<p><strong>Step 1 - </strong>
Multiply.<br> \((z^2-5z+2)\) by \(–3\).</p>
<p><strong>Step 2 - </strong>
Multiply. <br> \((z^2-5z+2)\) by \(z\).</p>
<p><strong>Step 3 - </strong>
Add like terms.</p>
<p>\(\underline{\underline{\;\;\;\;\;\;\;\;\;\;z^2-5z+2\\\;\;\;\;\times\;\;\;\;\;\;\;\;\;\;\;\;z-3}\\ \;\; \\ \;\;\;\;-3z^2+15z-6\\\;z^3-5z^2+2z}\\ \;\; \\z^3-8z^2+17z-6\)</p>
</div>