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<h1 align="center">Binomial Theorem</h1>
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<p>A <b>binomial</b> is a <a href="polynomials.html">polynomial</a> with two terms</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/binomial.svg" alt="Binomial" /></td>
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</tr>
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<tr align="center">
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<td>example of a binomial
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<br /> </td>
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</tr>
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</table>
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</div>
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<p class="large">What happens when we multiply a binomial by itself ... many times?</p>
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<div class="example">
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<h3>Example: <b>a+b</b></h3>
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<p><b>a+b</b> is a binomial (the two terms are <b>a</b> and <b>b</b>)</p>
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<p>Let us multiply <b>a+b</b> by itself using <a href="polynomials-multiplying.html">Polynomial Multiplication</a> :</p>
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<p class="center">(a+b)(a+b) = <b>a<sup>2</sup> + 2ab + b<sup>2</sup></b></p>
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<p>Now take that result and multiply by <b>a+b</b> again:</p>
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<p class="center"> (a<sup>2</sup> + 2ab + b<sup>2</sup>)(a+b) = <b>a<sup>3</sup> + 3a<sup>2</sup>b + 3ab<sup>2</sup> + b<sup>3</sup></b></p>
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<p>And again:</p>
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<p class="center">(a<sup>3</sup> + 3a<sup>2</sup>b + 3ab<sup>2</sup> + b<sup>3</sup>)(a+b) = <b>a<sup>4</sup> + 4a<sup>3</sup>b + 6a<sup>2</sup>b<sup>2</sup> + 4ab<sup>3</sup> + b<sup>4</sup></b> </p>
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</div>
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<p>The calculations get longer and longer as we go, but there is some kind of <b>pattern</b> developing.</p>
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<p>That pattern is summed up by the <b>Binomial Theorem</b>:</p>
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<div class="def">
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<p align="center"><img src="images/binomial-theorem.gif" alt="Binomial Theorem" width="244" height="64" />
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<br /> <span class="larger">The Binomial Theorem</span></p>
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</div>
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<p class="large">Don't worry ... it will all be explained! </p>
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<p>And you will learn lots of cool math symbols along the way.</p>
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<h2>Exponents</h2>
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<p><img style="float:right; margin: 0 0 5px 20px;" src="images/exponent-8-2.svg" alt="8 to the Power 2" /> </p>
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<p>First, a quick summary of <a href="../exponent.html">Exponents</a>. </p>
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<p>An exponent says <b>how many times </b>to use something in a multiplication.</p>
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<div class="example">
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<h3 class="Larger">Example: <b>8<sup>2</sup> = 8 × 8 = 64</b></h3> </div>
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<p>An exponent of <b>1</b> means just to have it appear once, so we get the original value:</p>
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<div class="example">
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<h3 class="Larger">Example: <b>8<sup>1</sup> = 8 </b></h3> </div>
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<p>An exponent of <b>0</b> means not to use it at all, and we have only 1:</p>
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<div class="example">
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<h3 class="Larger">Example: <b>8<sup>0</sup> = 1 </b></h3> </div>
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<h2>Exponents of (a+b)</h2>
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<p>Now on to the binomial. </p>
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<p align="center" class="larger">We will use the simple binomial <span class="large">a+b</span>, but it could be any binomial.</p>
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<p>Let us start with an exponent of <b>0</b> and build upwards.</p>
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<h3>Exponent of 0</h3>
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<p>When an exponent is 0, we get <b>1</b>:</p>
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<p align="center" class="large">(a+b)<sup>0</sup> = 1</p>
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<h3>Exponent of 1</h3>
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<p>When the exponent is 1, we get the original value, unchanged:</p>
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<p align="center" class="large">(a+b)<sup>1</sup> = a+b</p>
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<h3>Exponent of 2</h3>
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<p>An exponent of 2 means to multiply by itself (see <a href="polynomials-multiplying.html">how to multiply polynomials</a>):</p>
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<p align="center"><span class="large">(a+b)<sup>2</sup> = (a+b)(a+b) = a<sup>2</sup> + 2ab + b<sup>2</sup></span></p>
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<h3>Exponent of 3</h3>
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<p>For an exponent of 3 just multiply again:</p>
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<p align="center"><span class="large">(a+b)<sup>3</sup> = (a<sup>2</sup> + 2ab + b<sup>2</sup>)(a+b) = a<sup>3</sup> + 3a<sup>2</sup>b + 3ab<sup>2</sup> + b<sup>3</sup></span></p>
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<p> </p>
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<p><b>We have enough now to start talking about the pattern.</b></p>
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<h2>The Pattern</h2>
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<p>In the last result we got:</p>
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<p align="center"><span class="large">a<sup>3</sup> + 3a<sup>2</sup>b + 3ab<sup>2</sup> + b<sup>3</sup></span></p>
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<p>Now, notice the exponents of <span class="large">a</span>. They start at 3 and go down: 3, 2, 1, 0:</p>
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<p align="center"><img src="images/binomial-theorem-a-3210.gif" alt="a goes 3,2,1,0" width="287" height="66" /></p>
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<p>Likewise the exponents of <span class="large">b</span> go upwards: 0, 1, 2, 3:</p>
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<p align="center"><img src="images/binomial-theorem-b-0123.gif" alt="b goes 0,1,2,3" width="285" height="66" /></p>
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<p>If we number the terms 0 to <span class="number"><i>n</i></span>, we get this:</p>
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<table width="80%" border="0" align="center">
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<tr align="center">
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<td>k=0</td>
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<td>k=1</td>
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<td>k=2</td>
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<td>k=3</td>
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</tr>
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<tr align="center">
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<td><span class="large">a<sup>3</sup></span></td>
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<td><span class="large">a<sup>2</sup></span></td>
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<td><span class="large">a</span></td>
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<td><span class="large">1</span></td>
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</tr>
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<tr align="center">
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<td><span class="large">1</span></td>
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<td><span class="large">b</span></td>
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<td><span class="large">b<sup>2</sup></span></td>
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<td><span class="large">b<sup>3</sup></span></td>
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</tr>
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</table>
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<p>Which can be brought together into this:</p>
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<p align="center"><span class="large">a<sup>n-k</sup></span><span class="large">b<sup>k</sup></span></p>
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<p>How about an example to see how it works:</p>
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<div class="example">
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<h3>Example: When the exponent, <span class="number"><i>n</i></span>, is 3. </h3>
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<p>The terms are:</p>
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<table border="1" align="center" cellpadding="9">
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<tr align="center">
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<th>k=0:</th>
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<th>k=1:</th>
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<th>k=2:</th>
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<th>k=3:</th>
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</tr>
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<tr align="center">
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<td> <span class="larger"> a<sup>n-k</sup>b<sup>k</sup><br />
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= a<sup>3-0</sup>b<sup>0</sup><br />
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= <b>a<sup>3</sup></b></span></td>
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<td> <span class="larger"> a<sup>n-k</sup>b<sup>k</sup><br />
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= a<sup>3-1</sup>b<sup>1</sup><br />
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= <b>a<sup>2</sup>b</b></span></td>
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<td> <span class="larger">a<sup>n-k</sup>b<sup>k</sup><br />
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= a<sup>3-2</sup>b<sup>2</sup><br />
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= <b>ab<sup>2</sup></b></span></td>
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<td> <span class="larger">a<sup>n-k</sup>b<sup>k</sup><br />
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= a<sup>3-3</sup>b<sup>3</sup><br />
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= <b>b<sup>3</sup></b></span></td>
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</tr>
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</table>
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<p align="center">It works like magic!</p>
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<p align="center"></p>
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</div>
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<h2>Coefficients</h2>
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<div class="tbl">
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<div class="row"><span class="left">So far we have: </span><span class="right"><span class="large">a<sup>3</sup> + a<sup>2</sup>b + ab<sup>2</sup> + b<sup>3</sup></span></span></div>
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<div class="row"><span class="left">But we <b>really</b> need:</span><span class="right"><span class="large">a<sup>3</sup> + 3a<sup>2</sup>b + 3ab<sup>2</sup> + b<sup>3</sup></span></span></div>
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</div>
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<p>We are <b>missing the numbers</b> (which are called <i>coefficients</i>).</p>
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<p>Let's look at <b>all the results</b> we got before, from <span class="large">(a+b)<sup>0</sup></span> up to <span class="large">(a+b)<sup>3</sup></span>:</p>
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<p align="center"><img src="images/binomial-theorem-first4.svg" alt="1, a+b, a^2 + 2ab + b^2, a^3 + 3a^2b + 3ab^2 + b^3" /></p>
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<p>And now look at <b>just the coefficients</b> (with a "1" where a coefficient wasn't shown):</p>
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<p align="center"><img src="images/binomial-theorem-coeff.svg" alt="1, 1 1, 1 2 1, 1 3 3 1" /></p>
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<table border="0">
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<tr>
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<td>
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<p align="center" class="larger">They actually make <a href="../pascals-triangle.html">Pascal's Triangle</a>!</p>
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<h3>Each number is just the two numbers above it added together (except for the edges, which are all "1")</h3>
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<p align="right"> (Here I have highlighted that <b>1+3 = 4)</b></p>
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</td>
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<td><img src="../images/pascals-triangle-1.gif" width="206" height="210" alt="pascals triangle" /></td>
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</tr>
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</table>
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<p>Armed with this information let us try something new ... an <b>exponent of 4</b>:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td align="right"><span class="large">a</span> exponents go 4,3,2,1,0:</td>
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<td> </td>
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<td align="center"><span class="large">a<sup>4</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">a<sup>3</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">a<sup>2</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">a</span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">1</span></td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="right"><span class="large">b</span> exponents go 0,1,2,3,4:</td>
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<td> </td>
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<td align="center"><span class="large">a<sup>4</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">a<sup>3</sup>b</span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">a<sup>2</sup>b<sup>2</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">ab<sup>3</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">b<sup>4</sup></span></td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="right"><span class="large">coefficients</span> go 1,4,6,4,1:</td>
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<td> </td>
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<td align="center"><span class="large">a<sup>4</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">4a<sup>3</sup>b</span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">6a<sup>2</sup>b<sup>2</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">4ab<sup>3</sup></span></td>
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<td align="center"><span class="large"> +</span></td>
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<td align="center"><span class="large">b<sup>4</sup></span></td>
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<td align="center"><img src="../images/style/yes.svg" alt="yes" height="36" /></td>
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</tr>
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</table>
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</div>
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<p>And that is the correct answer (compare to the top of the page).</p>
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<p align="center" class="larger">We have success!</p>
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<p>We can now use that pattern for exponents of 5, 6, 7, ... 50, ... 112, ... you name it!</p>
|
|
<p> </p>
|
|
<div class="center80">
|
|
<p class="center">That pattern is the essence of the Binomial Theorem.</p>
|
|
<p class="center">Now you can take a break.</p>
|
|
<p class="center">When you come back see if you can work out <b>(a+b)<sup>5</sup></b> yourself.</p>
|
|
<p class="center">Answer (hover over): <span class="hide">a<sup>5</sup> + 5a<sup>4</sup>b + 10a<sup>3</sup>b<sup>2</sup> + 10a<sup>2</sup>b<sup>3</sup> + 5ab<sup>4</sup> + b<sup>5</sup></span></p>
|
|
</div>
|
|
<p> </p>
|
|
<h2>As a Formula</h2>
|
|
<p>Our next task is to write it all as a formula.</p>
|
|
<p>We already have the exponents figured out:</p>
|
|
<p align="center"><span class="large">a<sup>n-k</sup></span><span class="large">b<sup>k</sup></span></p>
|
|
<p>But how do we write a formula for <b>"find the coefficient from Pascal's Triangle"</b> ... ?</p>
|
|
<p>Well, there <b>is</b> such a formula:</p>
|
|
<p style="float:left; margin: 15px 30px 15px 0;"><img src="../data/images/binomial-n-choose-k.png" width="152" height="53" alt="binomial n choose k = n! / k!(n-k)!" /></p>
|
|
<p>It is commonly called "n choose k" because it is how many ways to choose k elements from a set of n.</p>
|
|
|
|
<p>The "!" means "<a href="../numbers/factorial.html">factorial</a>", for example 4! = 4×3×2×1 = 24 </p>
|
|
<p>You can read more at <a href="../combinatorics/combinations-permutations.html">Combinations and Permutations</a>.</p>
|
|
<table border="0" align="center">
|
|
<tr>
|
|
<td align="right">
|
|
<p>And it matches to Pascal's Triangle like this:</p>
|
|
<p>(Note how the top row is row zero
|
|
<br /> and also the leftmost column is zero!)</p>
|
|
</td>
|
|
<td><img src="../images/pascals-triangle-n-choose-k.gif" alt="Pascals Triangle Combinations" width="335" height="196" /></td>
|
|
</tr>
|
|
</table>
|
|
<div class="example">
|
|
<h3>Example: Row 4, term 2 in Pascal's Triangle is "6". </h3>
|
|
<p>Let's see if the formula works:</p>
|
|
<p align="center"><img src="../data/images/binomial-4-choose-2.gif" width="364" height="59" alt="binomial 4 choose 2 = 4! / 2!(4-2)!" /></p>
|
|
<p>Yes, it works! Try another value for yourself.</p>
|
|
</div>
|
|
<h2>Putting It All Together</h2>
|
|
<p>The last step is to put all the terms together into <b>one formula</b>. </p>
|
|
<p>But we are adding lots of terms together ... can that be done using one formula?</p>
|
|
<p>Yes! The handy <a href="sigma-notation.html">Sigma Notation</a> allows us to sum up as many terms as we want:</p>
|
|
<p align="center"><img src="images/sigma-notation.svg" alt="Sigma Notation" style="max-width:100%" />
|
|
<br> Sigma Notation</p>
|
|
<p>Now it can all go into one formula:</p>
|
|
<div class="def">
|
|
<p align="center"><img src="images/binomial-theorem.gif" alt="Binomial Theorem" width="244" height="64" />
|
|
<br /> <span class="larger">The Binomial Theorem</span></p>
|
|
</div>
|
|
<h2>Use It</h2>
|
|
<p>OK ... it won't make much sense without an example.</p>
|
|
<p>So let's try using it for <span class="number"><i>n</i> = 3</span> :</p>
|
|
<p align="center"><img src="images/binomial-theorem-4-2.gif" alt="Binomial Theorem" style="max-width:100%" /></p>
|
|
<p>BUT ... it is usually <b>much easier</b> just to remember the <b>patterns</b>:</p>
|
|
<ul>
|
|
<li>The first term's exponents start at <b>n and go down</b> </li>
|
|
<li>The second term's exponents start at <b>0 and go up</b> </li>
|
|
<li>Coefficients are from Pascal's Triangle, or by calculation using <span class="larger"><span class="intbl"><em>n!</em><strong>k!(n-k)!</strong></span></span></li>
|
|
</ul>
|
|
<p>Like this:</p>
|
|
<div class="example">
|
|
<h3>Example: What is (y+5)<sup>4</sup></h3>
|
|
<p> </p>
|
|
<table width="95%" border="0" align="center">
|
|
<tr align="center">
|
|
<td>Start with exponents:</td>
|
|
<td><b>y<sup>4</sup>5<sup>0</sup></b></td>
|
|
<td><b>y<sup>3</sup>5<sup>1</sup></b></td>
|
|
<td><b>y<sup>2</sup>5<sup>2</sup></b></td>
|
|
<td><b>y<sup>1</sup>5<sup>3</sup></b></td>
|
|
<td><b>y<sup>0</sup>5<sup>4</sup></b></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td>Include Coefficients:</td>
|
|
<td><b>1</b>y<sup>4</sup>5<sup>0</sup></td>
|
|
<td><b>4</b>y<sup>3</sup>5<sup>1</sup></td>
|
|
<td><b>6</b>y<sup>2</sup>5<sup>2</sup></td>
|
|
<td><b>4</b>y<sup>1</sup>5<sup>3</sup></td>
|
|
<td><b>1</b>y<sup>0</sup>5<sup>4</sup></td>
|
|
</tr>
|
|
</table>
|
|
|
|
<p> </p>
|
|
<p>Then write down the answer (including all calculations, such as 4×5, 6×5<sup>2</sup>, etc):</p>
|
|
<p align="center" class="large">(y+5)<sup>4</sup> = y<sup>4</sup> + 20y<sup>3</sup> + 150y<sup>2</sup> + 500y + 625</p>
|
|
</div>
|
|
<p> </p>
|
|
<p>We may also want to calculate just one term:</p>
|
|
<div class="example">
|
|
<h3>Example: What is the coefficient for x<sup>3</sup> in (2x+4)<sup>8</sup></h3>
|
|
<p>The <b>exponents</b> for x<sup>3</sup> are <b>8-5</b> (=3) for the "2x" and <b>5</b> for the "4":</p>
|
|
<p align="center" class="larger"> (2x)<sup>3</sup>4<sup>5</sup></p>
|
|
<p>(Why? Because:</p>
|
|
<table border="1" align="center">
|
|
<tr align="center">
|
|
<td><b>2x</b>:</td>
|
|
<td>8<sup></sup></td>
|
|
<td>7<sup></sup></td>
|
|
<td>6<sup></sup></td>
|
|
<td>5<sup></sup></td>
|
|
<td>4<sup></sup></td>
|
|
<td bgcolor="#CCFFCC">3</td>
|
|
<td>2</td>
|
|
<td>1</td>
|
|
<td>0</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td><b>4</b>:</td>
|
|
<td>0<sup></sup></td>
|
|
<td>1<sup></sup></td>
|
|
<td>2<sup></sup></td>
|
|
<td>3<sup></sup></td>
|
|
<td>4<sup></sup></td>
|
|
<td bgcolor="#CCFFCC">5</td>
|
|
<td>6</td>
|
|
<td>7</td>
|
|
<td>8</td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td> </td>
|
|
<td>(2x)<sup>8</sup>4<sup>0</sup><sup></sup></td>
|
|
<td>(2x)<sup>7</sup>4<sup>1</sup><sup></sup></td>
|
|
<td>(2x)<sup>6</sup>4<sup>2</sup><sup></sup></td>
|
|
<td>(2x)<sup>5</sup>4<sup>3</sup><sup></sup></td>
|
|
<td>(2x)<sup>4</sup>4<sup>4</sup><sup></sup></td>
|
|
<td bgcolor="#CCFFCC">(2x)<sup>3</sup>4<sup>5</sup></td>
|
|
<td>(2x)<sup>2</sup>4<sup>6</sup></td>
|
|
<td>(2x)<sup>1</sup>4<sup>7</sup></td>
|
|
<td>(2x)<sup>0</sup>4<sup>8</sup></td>
|
|
</tr>
|
|
</table>
|
|
<p>But we don't need to calculate all the other values if we only want one term.)</p>
|
|
<p> </p>
|
|
<p>And let's not forget "8 choose 5" ... we can use Pascal's Triangle, or calculate directly: </p>
|
|
<p class="center larger"><span class="intbl"><em>n!</em><strong>k!(n-k)!</strong></span> = <span class="intbl"><em>8!</em><strong>5!(8-5)!</strong></span> = <span class="intbl"><em>8!</em><strong>5!3!</strong></span> = <span class="intbl"><em>8×7×6</em><strong>3×2×1</strong></span> = 56</p>
|
|
|
|
<p>And we get:</p>
|
|
<p align="center"><span class="larger">56(2x)<sup>3</sup>4<sup>5</sup></span> </p>
|
|
<p>Which simplifies to:</p>
|
|
<p align="center" class="large"><span class="larger"><b>458752</b> x<sup>3</sup></span></p>
|
|
<p>A large coefficient, isn't it?</p>
|
|
</div>
|
|
<h2>Geometry</h2>
|
|
<p>The Binomial Theorem can be shown using Geometry:</p>
|
|
<p>In 2 dimensions, <span class="center"><b>(a+b)<sup>2</sup> = a<sup>2</sup> + 2ab + b<sup>2</sup></b></span></p>
|
|
<p class="center"><img src="images/binomial-squared.svg" alt="(a+b)^2 = a^2 + 2ab + b^2" style="max-width:100%" /></p>
|
|
<p> </p>
|
|
<p>In 3 dimensions, <b>(a+b)<sup>3</sup> = a<sup>3</sup> + 3a<sup>2</sup>b + 3ab<sup>2</sup> + b<sup>3</sup></b></p>
|
|
<p class="center"><img src="images/binomial-cubed.svg" alt="(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3" style="max-width:100%" /></p>
|
|
<p> </p>
|
|
<p>In 4 dimensions, <b>(a+b)<sup>4</sup> = a<sup>4</sup> + 4a<sup>3</sup>b + 6a<sup>2</sup>b<sup>2</sup> + 4ab<sup>3</sup> + b<sup>4</sup></b></p>
|
|
<p class="center"><i>(Sorry, I am not good at drawing in 4 dimensions!)</i></p>
|
|
<h2>Advanced Example</h2>
|
|
<p>And one last, most amazing, example:</p>
|
|
<div class="example">
|
|
<h3>Example: A formula for <b>e</b> (Euler's Number)</h3>
|
|
<p>We can use the Binomial Theorem to calculate <a href="../numbers/e-eulers-number.html">e (Euler's number)</a>.</p>
|
|
<p>e = <b>2.718281828459045...</b> (the digits go on forever without repeating)</p>
|
|
<p>It can be calculated using:</p>
|
|
<p align="center"><span class="large">(1 + 1/n)<sup>n</sup></span></p>
|
|
<p align="center">(It gets more accurate the higher the value of <b>n</b>)</p>
|
|
<p> </p>
|
|
<p>That formula is a <b>binomial</b>, right? So let's use the Binomial Theorem:</p>
|
|
<p align="center"><img src="images/binomial-theorem-e.gif" width="251" height="59" alt="(1 + 1/n)^n = Sigma k=0 to n of [ (n choose k) by 1^(n-k) by (1/n)^k ]" /></p>
|
|
<p>First, we can drop <span class="large">1<sup>n-k</sup></span> as it is always equal to 1:</p>
|
|
<p align="center"><img src="images/binomial-theorem-e2.gif" width="135" height="59" alt="Sigma k=0 to n of [ (n choose k) by (1/n)^k ]" /> </p>
|
|
<p>And, quite magically, most of what is left goes to <b>1</b> as n goes to infinity:</p>
|
|
<p align="center"><img src="images/binomial-theorem-e4.svg" alt="simplify steps in detail" style="max-width:100%" /></p>
|
|
<p>Which just leaves:</p>
|
|
<p align="center"><img src="images/binomial-theorem-e5.gif" width="317" height="111" alt="Sigma k=0 to infinity of 1/k! = 1 + 1 + 1/2 + 1/6 + 1/24 + ..." /></p>
|
|
<p> </p>
|
|
<p>With just those first few terms we get <span class="large">e ≈ 2.7083... </span></p>
|
|
<p>Try calculating more terms for a better approximation! (Try the <a href="../numbers/sigma-calculator.html">Sigma Calculator</a>)</p>
|
|
</div>
|
|
<p> </p>
|
|
<div class="questions">
|
|
<script type="text/javascript">
|
|
getQ(615, 1299, 616, 1300, 2468, 8334, 8335, 8336, 8337, 9038);
|
|
</script>
|
|
<br /> <a href="javascript:doQ(12)">Challenging 1</a> <a href="javascript:doQ(13)">Challenging 2</a> </div>
|
|
<h2>Isaac Newton</h2>
|
|
<p>As a footnote it is worth mentioning that around 1665 Sir Isaac Newton came up with a "general" version of the formula that is not limited to exponents of 0, 1, 2, .... I hope to write about that one day.</p>
|
|
<div class="related"><a href="polynomials.html">Polynomial</a> <a href="../exponent.html">Exponent</a> <a href="../pascals-triangle.html">Pascal's Triangle</a> <a href="sigma-notation.html">Sigma Notation</a> <a href="index.html">Algebra Index</a></div>
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