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202 lines
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<title>Polynomials - Long Division</title>
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<!-- #BeginEditable "Body" -->
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<h1 align="center">Polynomials - Long Division</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
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<br /> 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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<h2>Dividing</h2>
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<p style="float:left; margin: 0 40px 5px 0;"><img src="images/polynomial-division.svg" alt="polynomial division" /></p>
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<p>Polynomials can sometimes be divided using the simple methods shown on <a href="polynomials-dividing.html">Dividing Polynomials</a>. </p>
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<p>But sometimes it is better to use "Long Division" (a method similar to <a href="../long_division.html">Long Division for Numbers</a>)</p>
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<h2>Numerator and Denominator</h2>
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<p>We can give each polynomial a name: </p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/polynomial-numerator-denominator.svg" alt="polynomial numerator denominator" /></p>
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<ul>
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<li>the <b>top</b> polynomial is the <b>numerator</b></li>
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<li>the <b>bottom</b> polynomial is the <b>denominator</b> </li>
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</ul>
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<p>If you have trouble remembering, think <b>denominator</b> is <b>down-</b>ominator.</p>
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<h2>The Method</h2>
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<p class="large">Write it down neatly:</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/polynomial-long-division.svg" alt="polynomial long division" /></p>
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<ul>
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<li>the denominator goes first, </li>
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<li>then a ")", </li>
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<li>then the numerator with a line above</li>
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</ul>
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<p class="large"> </p>
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<p>Both polynomials should have the "higher order" terms first (those with the largest <a href="../exponent.html">exponents</a>, like the "2" in x<sup>2</sup>). </p>
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<p class="large">Then:</p>
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<div class="beach">
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<table width="100%" border="0">
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<tr>
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<td><img src="../images/style/rotate.gif" alt="repeat" width="86" height="94" /></td>
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<td>
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<br />
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<ul>
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<li><span class="large">Divide</span> the first term of the numerator by the first term of the denominator, and put that in the answer.</li>
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<li><span class="large">Multiply</span> the denominator by that answer, put that below the numerator</li>
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<li><span class="large">Subtract</span> to create a new polynomial</li>
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</ul>
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</td>
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</tr>
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<tr>
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<td> </td>
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<td><b>Repeat</b>, using the new polynomial</td>
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</tr>
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</table>
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</div>
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<p>It is easier to show with an example!</p>
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<p><span class="large">Example: </span></p>
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<p align="center"><span class="larger"><img src="images/polynomials-division-long-a.png" alt="poly long division" width="109" height="45" /></span></p>
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<p>Write it down neatly like below, then solve it step-by-step (press play):</p>
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<script type="text/javascript">
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playerMain("images/polynomials-division-long-a.txt", 600, 300);
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</script>
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<h3>Check the answer:</h3>
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<p>Multiply the answer by the bottom polynomial, we should get the top polynomial:</p>
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<p align="center"><span class="larger"><img src="images/polynomials-division-long-a-check.gif" alt="poly long division" width="305" height="55" /></span> <img src="../images/style/yes.svg" alt="yes" height="46" /></p>
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<h2>Remainders</h2>
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<p>The previous example worked perfectly, but that is not always so! Try this one:</p>
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<script type="text/javascript">
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playerMain("images/polynomials-division-long-b.txt", 610, 300);
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</script>
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<p>After dividing we were left with "2", this is the "remainder".</p>
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<p align="center" class="larger">The remainder is what is left over after dividing.</p>
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<p>But we still have an answer: put the <b>remainder divided by the bottom polynomial </b> as part of the answer, like this:</p>
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<p align="center"><img src="images/polynomials-division-long-b.png" alt="poly long div b" width="263" height="44" /></p>
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<h2>"Missing" Terms</h2>
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<p>There can be "missing terms" (example: there may be an x<sup>3</sup>, but no x<sup>2</sup>). In that case either leave gaps, or include the missing terms with a coefficient of zero.</p>
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<p class="large">Example: </p>
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<p align="center"><img src="images/polynomials-division-long-c.png" alt="poly long div c" width="151" height="45" /></p>
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<p>Write it down with "0" coefficients for the missing terms, then solve it normally (press play):</p>
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<script type="text/javascript">
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playerMain("images/polynomials-division-long-c.txt", 610, 320);
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</script>
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<p>See how we needed a space for "3x<sup>3</sup>" ? </p>
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<h2>More than One Variable</h2>
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<p>So far we have been dividing polynomials with only one variable (<b>x</b>), but we can handle polynomials with two or more variables (such as <b>x</b> and <b>y</b>) using the same method.</p>
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<p class="large">Example: </p>
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<p align="center" class="large"> <img src="images/polynomials-division-long-d.png" alt="poly long div d" width="253" height="48" align="absmiddle" /></p>
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<script type="text/javascript">
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playerMain("images/polynomials-division-long-d.txt", 600, 320);
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</script>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">
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getQ(352, 353, 461, 1105, 1106, 1107, 2413, 2414, 3890, 3891);
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</script> </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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