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207 lines
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<title>Multiplying Polynomials</title>
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<meta name="description" content="To multiply two polynomials multiply each term in one polynomial by each term in the other polynomial." />
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<h1 align="center">Multiplying Polynomials</h1>
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<p><span class="larger">A <a href="polynomials.html">polynomial</a> looks like this:</span> </p>
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<div class="beach">
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<table border="0" align="center" cellpadding="5">
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<tr align="center">
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<td><img src="images/polynomial-example.svg" alt="polynomial example" /></td>
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</tr>
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<tr align="center">
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<td>example of a polynomial<br />
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this one has 3 terms</td>
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</tr>
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</table>
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</div>
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<p>To multiply two polynomials: </p>
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<ul>
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<li>multiply <b>each term</b> in one polynomial by <b>each term</b> in the other polynomial</li>
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<li>add those answers together, and simplify if needed</li>
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</ul>
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<p>Let us look at the simplest cases first.</p>
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<h2>1 term × 1 term (monomial times monomial)</h2>
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<p>To multiply one term by another term, first multiply the <b>constants</b>, then multiply <b>each variable</b> together and combine the result, like this (press play):</p>
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<script type="text/javascript">playerMain("images/variables-exponents-a.txt",600,100);</script>
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<p><i>(Note: I used "·" to mean multiply. In Algebra we don't like to use "×" because it looks too much like the letter "x")</i></p>
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<p align="center"><b>For more about multiplying terms, read <a href="variables-exponents-multiply.html">Multiply and Divide Variables with Exponents</a></b></p>
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<h2>1 term × 2 terms (monomial times binomial)</h2>
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<p>Multiply the single term by each of the two terms, like this:</p>
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<script type="text/javascript">playerMain("images/polynomials-multiply-a.txt",600,150);</script>
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<h2>2 term × 1 terms (binomial times monomial)</h2>
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<p>Multiply each of the two terms by the single term, like this:</p>
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<script type="text/javascript">playerMain("images/polynomials-multiply-b.txt",600,100);</script>
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<p>(I did that one a bit faster by multiplying in my head before writing it down)</p>
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<h2>2 terms × 2 terms (binomial times binomial)</h2>
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<br />
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<table width="70%" border="0" align="center">
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<tr>
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<td>Each of the <b>two terms </b>in the first binomial ...</td>
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</tr>
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<tr>
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<td align="center">... is <b>multiplied by </b>...</td>
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</tr>
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<tr>
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<td align="right">... each of the <b>two terms </b>in the second binomial</td>
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</tr>
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</table>
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<p>That is 4 differrent multiplications ... Why?</p>
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<div class="beach">
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<table border="0" align="center">
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<tr>
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<td><p style="float:left; margin: 20px 10px 5px 0;"><img src="images/tennis.jpg" alt="tennis" width="119" height="89" /></p>
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<h3>Matching up Partners</h3>
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<p>Two friends (Alice and Betty) challenge <br>
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two other friends (Charles and David) to <br>
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individual tennis matches.</p>
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<p>How many matches does that make? </p>
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<ul>
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<li>Alice plays Charles, and then Alice plays David</li>
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<li>Then Betty plays Charles and then Betty plays David</li>
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</ul>
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<p>They could play in any order, so long as <b>each of the first two friends</b><br />
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gets to play <b>each of the second two friends</b>.</p></td>
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</tr>
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</table>
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</div>
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<p class="large">It is the same when we multiply binomials!</p>
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<p>Instead of Alice and Betty, let's just use <b>a</b> and <b>b</b>, and Charles and David can be <b>c</b> and <b>d</b>:</p>
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<script type="text/javascript">playerMain("images/polynomials-multiply-c.txt",600,120);</script>
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<p>We can multiply them in any order so long as <b>each of the first two terms</b> gets multiplied by <b>each of the second two terms</b>.</p>
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<p>But there is a handy way to help us remember to multiply each term called "<b>FOIL</b>".</p>
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<p>It stands for "<b>F</b>irsts, <b>O</b>uters, <b>I</b>nners, <b>L</b>asts":</p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/foil.svg" alt="foil" /></td>
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<td><ul>
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<li>Firsts: <b>ac</b></li>
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<li>Outers: <b>ad</b></li>
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<li>Inners: <b>bc</b></li>
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<li>Lasts: <b>bd</b></li>
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</ul></td>
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</tr>
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</table>
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<p>So you multiply the "Firsts" (the first terms of both polynomials), then the "Outers", etc.</p>
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<p>Let us try this on a more complicated example:</p>
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<script type="text/javascript">playerMain("images/polynomials-multiply-d.txt",620,130);</script>
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<h2>2 terms × 3 terms (binomial times trinomial)</h2>
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<p>"FOIL" won't work here, because there are more terms now. But just remember: </p>
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<p align="center" class="larger">Multiply each term in the first polynomial by each term in the second polynomial</p>
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<script type="text/javascript">playerMain("images/polynomials-multiply-e.txt",600,180);</script>
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<h2>Like Terms</h2>
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<p>And always remember to add <a href="like-terms.html">Like Terms</a>:</p>
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<div class="example">
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<h3>Example: (x + 2y)(3x − 4y + 5)</h3>
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<p align="center" class="larger">(x + 2y)(3x − 4y + 5) </p>
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<p align="center" class="larger">= 3x<sup>2</sup> <span class="hilite">− 4xy</span> + 5x <span class="hilite">+ 6xy</span> − 8y<sup>2</sup> + 10y </p>
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<p align="center" class="larger">= 3x<sup>2</sup> <span class="hilite">+ 2xy</span> + 5x − 8y<sup>2</sup> + 10y</p>
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<p>Note: <b>−4xy</b> and <b>6xy</b> are added because they are Like Terms.</p>
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<p>Also note: <b>6yx</b> means the same thing as <b>6xy</b></p>
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</div>
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<h2>Long Multiplication</h2>
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<p>You may also like to read about <a href="polynomials-multiplication-long.html">Polynomial Long Multiplication</a></p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(346, 1100, 347, 1101, 1102, 2411, 2412, 3885, 3886);</script>
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<a href="javascript:doQ(86)">Harder Question</a></div>
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<div class="related"> <a href="polynomials.html">Polynomials</a> <a href="introduction.html">Introduction to Algebra</a> <a href="definitions.html">Algebra - Basic Definitions</a> <a href="index.html">Algebra Index</a> </div>
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Copyright © 2018 MathsIsFun.com
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