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<title>Polynomials: Bounds on Zeros</title>
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<h1 align="center">Polynomials: Bounds on Zeros</h1>
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<p align="center"><i>A clever way to know where to search for roots.</i></p>
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<p>A <a href="polynomials.html">Polynomial</a> looks like this: </p>
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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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<p>A polynomial has <b>coefficients</b>:</p>
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<p align="center"><img src="images/polynomial-coefficients.svg" alt="polynomial coefficients" /><br>
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The terms are in order from highest to lowest exponent</p>
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<p>(Technically the 7 is a constant, but here it is easier to think of them all as coefficients.)</p>
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<p>A polynomial also has <b>roots</b>:</p>
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<p align="center"><img src="images/roots.svg" alt="polynomial coefficients" /></p>
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<div class="center80">
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<p align="center">A "root" (or "zero") is where the <b>polynomial is equal to zero</b>.</p>
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<p align="center">Example: <b>3x − 6 </b>equals<b> zero </b> when <b>x=2</b>, because 3(2)−6 = 6−6 = 0</p>
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</div>
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<h2>Where are the Roots (Zeros)?</h2>
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<p>It can sometimes be hard to find where the roots are! </p>
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<p align="center" class="larger">... where should we search ... how far left or right should we go?</p>
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<p>Here we will see a clever way to know where to search for all Real roots.</p>
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<p>And it just uses simple arithmetic!</p>
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<h2>Steps</h2>
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<p>First we prepare our data:</p>
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<ul>
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<div class="bigul">
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<li>The leading coefficient must be 1. If it is not, then divide every term of the polynomial by the leading coefficient</li>
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<li>Write down all the coefficients</li>
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<li>Then throw away the leading coefficient!</li>
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<li>Remove minus signs</li>
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<li>And we now have a list of values for the next step</li>
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</div>
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</ul>
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<p> </p>
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<p>Now we can calculate two different "bounds" using those values:</p>
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<ul>
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<div class="bigul">
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<li>Bound 1: The <b>largest value</b>, <b>plus 1</b></li>
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<li>Bound 2: The <b>sum of all values</b>, or <b>1</b>, whichever is larger</li>
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</div>
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</ul>
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<p>The <b>smallest</b> of those 2 bounds is our answer ...</p>
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<p align="center" class="larger">... all roots are within plus or minus of that!</p>
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<p> </p>
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<h2>Examples</h2>
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<div class="example">
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<h3>Example: x<sup>3</sup> + 2x<sup>2</sup> − 5x + 1</h3>
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<p>The leading coefficient is 1, so we can continue.</p>
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<p>The coefficients are: 1, 2, −5, 1</p>
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<p> Drop the leading coefficient, and remove any minus signs: <b>2, 5, 1</b></p>
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<ul>
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<li>Bound 1: the largest value is 5. Plus 1 = <b>6</b><br />
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</li>
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<li>Bound 2: adding all values is: 2+5+1 = <b>8</b></li>
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</ul>
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<p>The smallest bound is <b>6</b></p>
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<p align="center" class="larger">All Real roots are between <b>−6</b> and <b>+6</b></p>
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<p>So we can graph between −6 and 6 and find any Real roots. It is best to plot a little wider so we could see if a curve has roots <b>right at</b> −6 or 6:</p>
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<p align="center"><img src="images/polynomial-bounds1.gif" alt="polynomial bounds" width="242" height="158" /></p>
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<p>Now we can just <a href="../data/function-grapher3729.html?func1=x^3+2x^2-5x+1&xmin=-10&xmax=10&ymin=-6.17&ymax=7.17">zoom in to the graph</a> to get more accurate values for the roots</p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: 10x<sup>5</sup> + 2x<sup>3</sup> − x<sup>2</sup> − 3</h3>
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<p>the leading coefficient is 10, so we must divide all terms by 10:</p>
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<p align="center" class="larger">x<sup>5</sup> + 0.2x<sup>3</sup> − 0.1x<sup>2</sup> − 0.3</p>
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<p>The coefficients are: 1, 0.2, −0.1, −0.3<br />
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Drop the leading coefficient, and remove any minus signs: <b>0.2, 0.1, 0.3</b></p>
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<ul>
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<li>Bound 1: the largest value is 0.3. Plus 1 = <b>1.3</b></li>
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<li> Bound 2: adding all values is: 0.2+0.1+0.3 = <b>0.6</b>, which is less than 1, so the answer is <b>1</b></li>
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</ul>
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<p>The smallest is <b>1</b>.</p>
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<p align="center" class="larger">All Real roots are between <b>−1</b> and <b>+1</b></p>
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<p>I will leave the <a href="../data/function-grapher.html">graphing</a> to you.</p>
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</div>
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<h2>Notes</h2>
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<p>"Bound 1" and "Bound 2" are not the only ways to find the bounds of the roots, but they are easy to use!</p>
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<p>Also Note: Graphing polynomials can only find <a href="../numbers/real-numbers.html">Real</a> roots, but there can also be <a href="../numbers/complex-numbers.html">Complex</a> roots.</p>
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<p> </p>
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