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144 lines
6.6 KiB
HTML
144 lines
6.6 KiB
HTML
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<head>
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<title>General Form of a Polynomial</title>
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<h1 align="center">General Form of a Polynomial</h1>
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<p class="larger">A <a href="polynomials.html">polynomial</a> with one variable looks like this: </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-1var-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>But how do we talk about <b>general</b> polynomials? Ones that may have lots of terms?</p>
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<h2>General Form</h2>
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<p>A general polynomial (of one variable) could have <b>any number of terms</b>:</p>
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<div class="tbl">
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<div class="row"><span class="left">Degree 2 (Quadratic) can have letters a,b,c:</span><span class="right">ax<sup>2</sup> + bx + c</span></div>
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<div class="row"><span class="left">Degree 3 (Cubic) can have letters a,b,c,d:</span><span class="right">ax<sup>3</sup> + bx<sup>2</sup> + cx + d</span></div>
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<div class="row"><span class="left">...</span><span class="right">...</span></div>
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<div class="row"><span class="left">But for Degree "n" <b>letters won't work</b>:</span><span class="right">ax<sup>n</sup> + bx<sup>n-1</sup> + ... + <b>?</b>x + <b>?</b></span></div>
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</div>
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<p class="center large">The trouble is, we don't know what letters to end on!</p>
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<table border="0" align="center">
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<tr>
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<td>So instead of "a, b, c, ..." we use the letter "a" with a <b>little number</b> next to it, which says which term it belongs to:</td>
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<td><img src="images/polynomial-general-term.gif" alt="polynomial general term" width="108" height="67" /></td>
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</tr>
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</table>
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<p>So for the <b>general</b> case, we use this style: </p>
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<p align="center"><img src="images/polynomial-general-form.gif" alt="polynomial general form" width="387" height="30" /></p>
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<p>And now we can say:</p>
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<div class="bigul">
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<ul>
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<li><b>a<sub>n</sub></b> is the coefficient (the number we multiply by) for <b>x<sup>n</sup></b>, </li>
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<li><b>a<sub>n-1</sub></b> is the coefficient for <b>x<sup>n-1</sup></b>, </li>
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<li>... etc, down to ...</li>
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<li><b>a<sub>1</sub></b> which is the coefficient for <b>x</b> (because x<sup>1</sup> = x), and </li>
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<li><b>a<sub>0</sub></b> which is the constant term (because x<sup>0</sup> = 1). </li>
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</ul>
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</div>
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<div class="example">
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<h3>Example: 9x<sup>4</sup> + 5x<sup>2</sup> - x + 7</h3>
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<ul>
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<li>a<sub>4</sub> = <b>9</b></li>
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<li>a<sub>3</sub> = <b>0</b> (there is no x<sup>3</sup> term)</li>
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<li>a<sub>2</sub> = <b>5</b></li>
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<li>a<sub>1</sub> = <b>-1</b></li>
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<li>a<sub>0</sub> = <b>7</b></li>
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</ul>
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</div>
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<p>Note also:</p>
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<ul>
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<div class="bigul">
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<li>The <a href="degree-expression.html">Degree</a> of the polynomial is <b>n</b></li>
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<li> <span class="larger"><b>a<sub>n</sub></b></span> is the coefficient of the highest term <b>x<sup>n</sup></b></li>
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<li><span class="larger"><b>a<sub>n</sub></b></span> is not equal to zero (otherwise no <b>x<sup>n</sup></b> term)</li>
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<li><span class="larger"><b>a<sub>n</sub></b> is always a</span> Real Number</li>
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<li><b>n</b> can be 0, 1, 2, and so on, but not infinity</li>
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</div>
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</ul>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(1126, 4018, 9030, 9031, 9032, 9033, 9034, 9035, 1127, 4019);</script>
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<div class="related"><a href="index.html">Algebra Index</a></div>
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