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145 lines
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HTML
145 lines
6.6 KiB
HTML
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<title>Difference of Two Cubes</title>
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<h1 align="center">Difference of Two Cubes</h1>
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<p align="center"><i>There is a <b>special case</b> when multiplying polynomials that produces this: <b>a<sup>3</sup> − b</b></i><b><i><sup>3</sup></i></b></p>
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<h2>Polynomials</h2>
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<p>A <a href="polynomials.html">polynomial</a> 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 2x^4+6x-5" /></td>
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</tr>
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<tr align="center">
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<td>example of a polynomial<br /></td>
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</tr>
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</table>
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</div>
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<h2>Difference of Two Cubes</h2>
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<p>The <b>Difference of Two Cubes</b> is a special case of <a href="polynomials-multiplying.html">multiplying polynomials</a> : </p>
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<div class="def">
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<p align="center" class="large">(a−b)(a<sup>2</sup>+ab+b<sup>2</sup>) = a<sup>3</sup> − b<sup>3</sup></p>
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</div>
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<p>It comes up sometimes when solving things, so is worth remembering.</p>
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<p>And this is why it works out so simply (press play):</p>
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<script type="text/javascript">playerMain("images/difference-cubes.txt",600,200);</script>
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<h2>Example from Geometry:</h2>
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<p>Take two cubes of lengths x and y:</p>
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<p><img src="images/polynomial-cubes-a.gif" alt="polynomial cubes" width="248" height="127" /></p>
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<p>The larger "x" cube can be split into four smaller boxes (cuboids), with box <b>A being a cube of size "y"</b>:<br />
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</p>
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<p><img src="images/polynomial-cubes-b.gif" alt="polynomial cubes" width="657" height="133" /></p>
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<p>The volumes of these boxes are:</p>
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<ul>
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<li>A = y<sup>3</sup></li>
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<li>B = x<sup>2</sup>(x − y)</li>
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<li>C = xy(x − y)</li>
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<li>D = y<sup>2</sup>(x − y)</li>
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</ul>
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<p>But together, A, B, C and D make up the larger cube that has volume x<sup>3</sup>: <br />
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</p>
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<table border="0" align="center">
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<tr>
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<td align="right" class="larger">x<sup>3</sup></td>
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<td class="larger"> = </td>
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<td class="larger"> y<sup>3</sup> + x<sup>2</sup>(x − y) + xy(x − y) + y<sup>2</sup>(x − y)</td>
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</tr>
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<tr>
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<td align="right"><span class="larger">x<sup>3</sup> − y<sup>3</sup></span></td>
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<td><span class="larger"> = </span></td>
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<td class="larger">x<sup>2</sup>(x − y) + xy(x − y) + y<sup>2</sup>(x − y)</td>
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</tr>
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<tr>
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<td align="right"><span class="larger">x<sup>3</sup> − y<sup>3</sup></span></td>
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<td><span class="larger"> = </span></td>
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<td class="larger"> (x − y)(x<sup>2</sup> + xy + y<sup>2</sup>)</td>
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</tr>
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</table>
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<p>Hey! We ended up with the same formula! Thank goodness.</p>
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<h2>Sum of Two Cubes</h2>
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<p>There is also the "Sum of Two Cubes"</p>
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<p>By changing the sign of <span class="large">b</span> in each case we get:</p>
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<div class="def">
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<p align="center" class="large">(a+b)(a<sup>2</sup>−ab+b<sup>2</sup>) = a<sup>3</sup> + b<sup>3</sup> </p>
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</div>
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<p>(note the minus in front of "ab" also)</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(464, 465, 1112, 1113, 4007, 4008, 2260, 4009, 2261, 4010);</script>
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<div class="related"><a href="index.html">Algebra Index</a></div>
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