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<title>Polynomials: The Rule of Signs</title>
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<h1 align="center">Polynomials: The Rule of Signs</h1>
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<p align="center"><i>A special way of telling how many positive and negative roots a polynomial has</i>.</p>
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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 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>Polynomials have "roots" (zeros), where they are <b>equal to 0</b>:</p>
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<p align="center"><img src="images/polynomial-roots.gif" alt="polynomial roots" width="137" height="152" /><br />
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Roots are at <b>x=2</b> and <b>x=4</b><br />
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It has 2 roots, and <b>both are positive</b> (+2 and +4)</p>
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<p>Sometimes we may not know <b>where</b> the roots are, but we can say how many are positive or negative ...</p>
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<p align="center" class="large"> ... just by counting how many times the sign changes <br />
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(from plus to minus, or minus to plus)</p>
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<p> </p>
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<p>Let me show you with an example:</p>
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<div class="center80">
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<p align="center" class="large">Example: 4x + x<sup>2 </sup>− 3x<sup>5 </sup>− 2</p>
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</div>
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<h2>How Many of The Roots are Positive?</h2>
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<p>First, rewrite the polynomial <b>from highest to lowest exponent</b> (ignore any "zero" terms, so it does not matter that <span class="larger">x<b><sup>4</sup></b></span> and <span class="larger">x<b><sup>3</sup></b></span> are missing):</p>
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<p align="center" class="larger">−3x<b><sup>5</sup></b> + x<b><sup>2</sup></b> + 4x − 2</p>
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<p>Then, count how many times there is a <b>change of sign</b> (from plus to minus, or minus to plus):</p>
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<p align="center"><img src="images/polynomial-rule-signs1.svg" alt="Rule of Signs" /></p>
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<div class="center80">
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<p>The number of <b>sign changes</b> is the maximum number of <b>positive roots</b></p>
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</div>
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<p>There are <b>2 changes</b> in sign, so there are <b>at most 2 positive roots</b> (maybe less).</p>
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<p align="center" class="larger">So there could be <b>2, or 1, or 0 positive roots</b> ?</p>
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<p>But actually there won't be just 1 positive root ... read on ...</p>
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<h2>Complex Roots</h2>
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<p>There <b>might also be</b> complex roots.</p>
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<div class="def">
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<p align="center">A <a href="../numbers/complex-numbers.html">Complex Number</a> is a combination of a <a href="../numbers/real-numbers.html">Real Number</a> and an <a href="../numbers/imaginary-numbers.html">Imaginary Number</a></p>
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</div>
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<p align="center"><img src="../numbers/images/complex-number.svg" alt="Complex Number" /></p>
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<p>But ... </p>
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<p align="center" class="larger">Complex Roots <b>always come in pairs</b>!</p>
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<p align="center"><img src="images/complex-conjugate-pair.gif" alt="Complex Conjugate Pairs" width="227" height="171" /></p>
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<p>Always in pairs? Yes. So we either get: </p>
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<ul>
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<li><b>no</b> complex roots,</li>
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<li><b>2</b> complex roots,</li>
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<li><b>4</b> complex roots,</li>
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<li>etc</li>
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</ul>
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<h2>Improving the Number of Positive Roots</h2>
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<p>Having complex roots will <b>reduce the number of positive roots</b> by 2 (or by 4, or 6, ... etc), in other words by an <b>even number</b>.</p>
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<p>So in our example from before, instead of <b>2</b> positive roots there might be <b>0</b> positive roots:</p>
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<p align="center" class="larger">Number of Positive Roots is <b>2</b>, or <b>0</b></p>
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<p>This is the general rule:</p>
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<div class="center80">
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<p>The number of positive roots equals <b>the number of sign changes</b>, or a value less than that by some <b>multiple of 2</b></p>
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</div><p> </p>
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<div class="example">
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<p>Example: If the maximum number of positive roots was <b>5</b>, then there could be <b>5</b>, or <b>3</b> or <b>1</b> positive roots.</p>
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</div>
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<h2>How Many of The Roots are Negative?</h2>
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<p>By doing a similar calculation we can find out how many roots are <b>negative</b> ...</p>
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<p align="center">... but first we need to <b>put "−x" in place of "x"</b>, like this:</p>
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<p align="center"><img src="images/polynomial-rule-signs2.svg" alt="Rule of Signs" /></p>
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<p>And then we need to work out the signs:</p>
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<ul>
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<li><span class="hilite">−</span>3(−x)<sup>5</sup> becomes <span class="hi">+</span>3x<sup>5</sup></li>
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<li><span class="hi">+</span>(−x)<sup>2</sup> becomes <span class="hi">+</span>x<sup>2</sup> (no change in sign)</li>
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<li><span class="hi">+</span>4(−x) becomes <span class="hi">−</span>4x</li>
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</ul>
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<p>So we get:</p>
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<p align="center"><span class="larger">+3x<b><sup>5</sup></b> + x<b><sup>2</sup></b> − 4x − 2</span></p>
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<p><i>The trick is that only the <b>odd exponents</b>, like 1,3,5, etc will reverse their sign.</i></p>
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<p>Now we just count the changes like before:</p>
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<p align="center"><img src="images/polynomial-rule-signs3.svg" alt="Rule of Signs" /></p>
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<p align="center">One change only, so there <b>is 1 negative root</b>.<br />
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</p>
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<h3>But remember to reduce it because there may be Complex Roots! </h3>
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<p>But hang on ... we can only reduce it by an even number ... and 1 cannot be reduced any further ... so <b>1 negative root</b> is the only choice.</p>
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<h2>Total Number of Roots</h2>
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<p>On the page <a href="fundamental-theorem-algebra.html">Fundamental Theorem of Algebra</a> we explain that a polynomial will have <b>exactly as many roots as its degree</b> (the degree is the highest exponent of the polynomial).</p>
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<p align="center"><img src="images/polynomial-rule-signs4.svg" alt="Rule of Signs" /></p>
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<p>So we know one more thing: the degree is 5 so <b>there are 5 roots in total</b>.<br />
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</p>
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<h2>What we Know</h2>
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<p>OK, we have gathered lots of info. We know all this:</p>
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<ul>
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<li>positive roots: <b>2</b>, or <b>0 </b></li>
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<li>negative roots: <b>1</b> </li>
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<li>total number of roots: <b>5</b> </li>
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</ul>
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<p>So, after a little thought, the overall result is:</p>
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<ul>
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<li><b>5</b> roots: <b>2</b> positive, <b>1</b> negative, <b>2</b> complex (one pair), <b>or</b></li>
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<li><b>5</b> roots: <b>0</b> positive, <b>1</b> negative, <b>4</b> complex (two pairs) </li>
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</ul>
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<p align="center"><b>And we managed to figure all that out just based on the signs and exponents!</b></p>
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<h2>Must Have a Constant Term</h2>
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<p>One last important point:</p>
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<div class="center80">
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<p>Before using the Rule of Signs the polynomial <b>must have a constant term</b> (like "+2" or "−5")</p>
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</div>
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<p>If it doesn't, then just factor out <b>x</b> until it does.</p>
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<div class="example">
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<h3>Example: 2x<sup>4</sup> + 3x<sup>2</sup> − 4x</h3>
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<p>No constant term! So factor out "x": </p>
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<p align="center" class="larger">x(2x<sup>3</sup> + 3x − 4)</p>
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<p>This means that <b>x=0</b> is one of the roots.</p>
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<p>Now do the "Rule of Signs" for: </p>
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<p align="center"><span class="larger">2x<sup>3</sup> + 3x<sup> </sup> − 4</span></p>
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<p>Count the sign changes for positive roots: </p>
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<p align="center"><img src="images/polynomial-rule-signs5.svg" alt="Rule of Signs" /><br />
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There is just one sign change,<br />
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So there is <b>1 positive root</b></p>
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<p>And the negative case (after flipping signs of odd-valued exponents): </p>
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<p align="center"><img src="images/polynomial-rule-signs6.svg" alt="Rule of Signs" /><br />
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There are no sign changes,<br />
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So there are <b>no negative roots</b></p>
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<p align="center"></p>
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<p>The degree is 3, so we expect 3 roots. There is only one possible combination: </p>
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<ul>
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<li>3 roots: 1 positive, 0 negative and 2 complex</li>
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</ul>
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<p> </p>
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<p>And now, back to the original question:</p>
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<p align="center" class="larger">2x<sup>4</sup> + 3x<sup>2</sup> − 4x</p>
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<p>Will have:</p>
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<ul>
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<li>4 roots: 1 zero, 1 positive, 0 negative and 2 complex</li>
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</ul>
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</div>
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<p> </p><div class="questions">
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<script type="text/javascript">getQ(489, 490, 1130, 1131, 2420, 2421, 4023, 4024, 4025, 4026);</script>
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</div>
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<div class="words">
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<p>Historical Note: The Rule of Signs was first described by René Descartes in 1637, and is sometimes called <b>Descartes' Rule of Signs</b>.</p>
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</div>
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Copyright © 2017 MathsIsFun.com
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