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173 lines
8.6 KiB
HTML
173 lines
8.6 KiB
HTML
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<head>
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<!-- #BeginEditable "doctitle" -->
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<title>Adding and Subtracting Polynomials</title>
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<meta name="description" content="To add polynomials we simply add any like terms together ... so what is a like term?" />
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Adding and Subtracting Polynomials</h1>
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<p><span class="larger">A polynomial 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 add polynomials we simply add any <b>like terms</b> together ... so what is a like term?</p>
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<h2>Like Terms</h2>
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<p><a href="like-terms.html">Like Terms</a> are <b>terms</b> whose variables (and their <a href="../exponent.html">exponents</a> such as the 2 in x<sup>2</sup>) are the same.</p>
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<p> In other words, terms that are "like" each other. </p>
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<p>Note: the <b>coefficients</b> (the numbers you multiply by, such as "5" in 5x) can be different.</p>
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<div class="example">
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<h3>Example:</h3>
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<table border="0" align="center">
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<tr align="center">
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<td width="80" class="larger" >7<b>x</b></td>
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<td width="80" class="larger" ><b>x</b></td>
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<td width="80" class="larger" > -2<b>x</b></td>
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<td width="80" class="larger" ><span class="times">π</span><b>x</b></td>
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</tr>
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</table>
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<p>are all <b>like terms</b> because the variables are all <b>x</b></p>
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</div>
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<div class="example">
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<h3>Example:</h3>
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<table border="0" align="center">
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<tr align="center">
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<td width="90" class="larger" >(1/3)<b>xy<sup>2</sup></b></td>
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<td width="90" class="larger" >-2<b>xy<sup>2</sup></b></td>
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<td width="90" class="larger" >6<b>xy<sup>2</sup></b></td>
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<td width="90" class="larger" ><b>xy<sup>2</sup></b>/2</td>
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</tr>
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</table>
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<p>are all <b>like terms</b> because the variables are all <b>xy<sup>2</sup></b></p>
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</div>
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<div class="example">
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<h3>Example: These are <b>NOT</b> like terms because the variables and/or their exponents are different:</h3>
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<table border="0" align="center">
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<tr align="center">
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<td width="90" class="larger" >2<b style="color:blue;">x</b></td>
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<td width="90" class="larger" >2<b style="color:red;">x<sup>2</sup></b></td>
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<td width="90" class="larger" >2<b style="color:green;">y</b></td>
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<td width="90" class="larger" >2<b style="color:orange;">xy</b></td>
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</tr>
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</table>
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<div style="background-color:#ffaaaa"></div>
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</div>
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<h2>Adding Polynomials</h2>
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<p>Two Steps:</p>
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<ul>
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<li>Place <b>like terms</b> together</li>
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<li>Add the like terms</li>
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</ul>
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<div class="example">
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<p>Example: Add <b>2x<sup>2</sup> + 6x + 5</b> and <b>3x<sup>2</sup> - 2x - 1</b></p>
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<p> </p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">2x<sup>2</sup> + 6x + 5 + 3x<sup>2</sup> − 2x − 1</span></div>
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<div class="row"><span class="left">Place like terms together:</span><span class="right">2x<sup>2</sup>+3x<sup>2</sup> + 6x−2x + 5−1</span></div>
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<div class="row"><span class="left">Which is:</span><span class="right">(2+3)x<sup>2</sup> + (6−2)x + (5−1)</span></div>
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<div class="row"><span class="left">Add the like terms:</span><span class="right">5x<sup>2</sup> + 4x + 4</span></div>
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</div>
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</div>
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<p>Here is an animated example:</p>
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<script type="text/javascript">playerMain("images/polynomials-add-a.txt",600,210);</script>
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<p><i>(Note: there was no "like term" for the -7 in the other polynomial, so we didn't have to add anything to it.</i>)</p>
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<h2>Adding in Columns</h2>
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<p>We can also add them in columns like this:</p>
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<script type="text/javascript">playerMain("images/polynomials-add-b.txt",600,210);</script>
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<h2>Adding Several Polynomials</h2>
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<p>We can add several polynomials together like that.</p>
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<div class="example">
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<p>Example: Add <b>(2x<sup>2</sup> + 6y + 3xy)</b> , <b>(3x<sup>2</sup> - 5xy - x)</b> and <b>(6xy + 5)</b> </p>
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<p>Line them up in columns and add:</p>
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<p class="mono">2x<sup>2</sup> + 6y + 3xy<br />
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3x<sup>2</sup> - 5xy - x<br />
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<u> 6xy + 5</u></p>
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<p class="mono">5x<sup>2</sup> + 6y + 4xy - x + 5</p>
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</div>
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<p>Using columns helps us to match the correct terms together in a complicated sum.</p>
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<h2>Subtracting Polynomials</h2>
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<p>To subtract Polynomials, first <b>reverse the sign of each term</b> we are subtracting (in other words turn "+" into "-", and "-" into "+"), <b>then add</b> as usual. </p>
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<p>Like this:</p>
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<script type="text/javascript">playerMain("images/polynomials-add-c.txt",600,300);</script>
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<p><i>Note: After subtracting 2xy from 2xy we ended up with 0, so there is no need to mention the "xy" term any more.</i></p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(344, 345, 3186, 3187, 3883, 3884, 1098, 2094, 1099, 2095);</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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