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<title>Solving Polynomials</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" --> <h1 align="center">Solving Polynomials</h1>
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<span class="larger">A <a href="polynomials.html">polynomial</a> looks like this:</span>
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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 /></td>
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</tr>
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</table>
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</div>
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<h2>Solving</h2>
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<p>"Solving" means finding the "roots" ... </p>
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<p class="center">... a "root" (or "zero") is where the function <b>is equal to zero</b>: </p>
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<p class="center"><img src="images/inequality-graph-function.svg" alt="Graph of Inequality" /></p>
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<p class="center">In between the roots the function is either entirely above, <br />
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or entirely below, the x-axis</p>
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<div class="example">
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<h3>Example: −2 and 2 are the roots of the function x<sup>2</sup> − 4</h3>
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<p class="center"><img src="images/root-function.svg" alt="roots (zeros) of x^2-4"></p>
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<p>Let's check:</p>
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<ul>
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<li>when x = −2, then x<sup>2</sup> − 4 = (−2)<sup>2</sup> − 4 = 4 − 4 = <b>0</b></li>
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<li>when x = 2, then x<sup>2</sup> − 4 = 2<sup>2</sup> − 4 = 4 − 4 = <b>0</b></li>
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</ul>
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</div><p> </p>
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<p class="center larger"><b>How</b> do we solve polynomials? That depends on the <b>Degree</b>!</p>
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<h2>Degree</h2>
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<p>The <b>first step</b> in solving a polynomial is to find its degree.</p>
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<p>The <a href="degree-expression.html">Degree</a> of a Polynomial with one variable is ...</p>
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<p align="center">... the <a href="../exponent.html">largest exponent</a> of that variable.</p>
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<h3 align="center"><img src="images/degree-example-a.svg" alt="polynomial" /></h3>
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<p>When we know the degree we can also give the polynomial a name:</p>
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<div class="simple">
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<table border="0" align="center" cellpadding="0" cellspacing="0">
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<tr align="center" class="larger">
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<th>Degree</th>
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<th>Name</th>
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<th>Example</th>
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<th>Graph Looks Like</th>
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</tr>
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<tr align="center" class="larger">
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<td height="50" align="center">0</td>
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<td height="50">Constant</td>
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<td height="50" class="large">7</td>
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<td rowspan="4"><img src="images/polynomial-degree-graphs.svg" alt="polynomial degree graphs" /></td>
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</tr>
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<tr align="center" class="larger">
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<td height="50" align="center">1</td>
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<td height="50"><a href="linear-equations.html">Linear</a></td>
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<td height="50" class="large">4x+3</td>
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</tr>
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<tr align="center" class="larger">
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<td height="50" align="center">2</td>
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<td height="50"><a href="quadratic-equation.html">Quadratic</a></td>
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<td height="50" class="large">x<sup>2</sup>−3x+2</td>
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</tr>
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<tr align="center" class="larger">
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<td height="50" align="center">3</td>
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<td height="50">Cubic </td>
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<td height="50" class="large">2x<sup>3</sup>−5x<sup>2</sup></td>
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</tr>
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<tr align="center" class="larger">
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<td height="50" align="center">4</td>
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<td height="50">Quartic</td>
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<td height="50" class="large">x<sup>4</sup>+3x−2</td>
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<td>...</td>
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</tr>
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<tr align="center" class="larger">
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<td height="50" align="center">etc</td>
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<td height="50">...</td>
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<td height="50">...</td>
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<td>...</td>
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</tr>
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</table>
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</div>
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<h2>How To Solve</h2>
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<p>So now we know the degree, how to solve?</p>
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<div class="bigul">
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<ul>
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<li>Read how to solve <a href="linear-equations.html">Linear Polynomials</a> (Degree 1) using simple algebra.</li>
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<li>Read how to solve <a href="quadratic-equation.html">Quadratic Polynomials</a> (Degree 2) with a little work,</li>
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<li>It can be hard to solve Cubic (degree 3) and Quartic (degree 4) equations,</li>
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<li>And beyond that it <b>can be impossible</b> to solve polynomials directly.</li>
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</ul>
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</div>
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<p>So what do we do with ones we can't solve? Try to solve them a piece at a time!</p>
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<div class="center80">
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<p>If we find one root, we can then <b>reduce the polynomial by one degree</b> (example later) and this may be enough to solve the whole polynomial.</p>
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</div>
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<p>Here are some main ways to find roots.</p>
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<h3>1. Basic Algebra</h3>
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<p>We may be able to solve using basic algebra:</p>
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<div class="example">
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<h3>Example: <b>2x+1</b> </h3>
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<p><b>2x+1</b> is a linear polynomial:</p>
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<p align="center"><img src="images/graph-line.svg" alt="line on a graph" /></p>
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<p align="center">The graph of <b>y = 2x+1</b> is a straight line</p>
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<p>It is linear so there is one root. </p>
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<p>Use Algebra to solve:</p>
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<p align="center">A "root" is when y is zero: <span class="larger">2x+1 = 0</span></p>
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<p align="center">Subtract 1 from both sides: <span class="larger">2x = −1</span></p>
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<p align="center">Divide both sides by 2: <span class="larger">x = −1/2</span></p>
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<p>And that is the solution: </p>
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<p align="center" class="large"> x = −1/2</p>
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<p>(You can also see this on the graph)</p>
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</div>
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<p>We can also solve <a href="quadratic-equation.html">Quadratic Polynomials</a> using basic algebra (read that page for an explanation).</p>
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<p> </p>
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<h3>2. By experience, or simply guesswork. </h3>
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<p>It is always a good idea to see if we can do simple factoring:</p>
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<div class="example">
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<h3>Example: x<sup>3</sup>+2x<sup>2</sup>−x</h3>
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<p>This is cubic ... but wait ... we can factor out "x":</p>
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<p align="center" class="larger">x<sup>3</sup>+2x<sup>2</sup>−x = x(x<sup>2</sup>+2x−1)</p>
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<p>Now we have one root (x=0) and what is left is quadratic, which we can solve exactly.</p>
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</div>
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Or we may notice a familiar pattern:
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<div class="example">
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<h3>Example: x<sup>3</sup>−8</h3>
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<p>Again this is cubic ... but it is also the "<a href="polynomials-difference-two-cubes.html">difference of two cubes</a>":</p>
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<p align="center" class="larger">x<sup>3</sup>−8 = x<sup>3</sup>−2<sup>3</sup></p>
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<p>And so we can turn it into this:</p>
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<p align="center" class="larger">x<sup>3</sup>−8 = (x−2)(x<sup>2</sup>+2x+4) </p>
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<p>There is a root at x=2, because:</p>
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<p align="center"> <span class="larger">(2−2)(2<sup>2</sup>+2×2+4) = <b>(0)</b>(2<sup>2</sup>+2×2+4)</span></p>
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<p>And we can then solve the quadratic <span class="larger">x<sup>2</sup>+2x+4</span> and we are done</p>
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</div>
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<p> </p>
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<h3>3. Graphically. </h3>
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<p>Graph the polynomial and see where it crosses the x-axis.</p>
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<table width="80%" border="0" align="center">
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<tr>
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<td><a href="../data/function-grapher.html"><img src="../data/images/function-grapher-sm.gif" alt="Function Grapher" width="86" height="55" /></a></td>
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<td> </td>
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<td>We can enter the polynomial into the <a href="../data/function-grapher.html">Function Grapher</a>, and then zoom in to find where it crosses the x-axis.</td>
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</tr>
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</table>
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<p>Graphing is a good way to find approximate answers, and we may also get lucky and discover an exact answer.</p>
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<div class="center80">
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<p class="larger">Caution: before you jump in and graph it, you should really know <a href="polynomials-behave.html">How Polynomials Behave</a>, so you find all the possible answers!</p>
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</div>
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<h2>Factors</h2>
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<p><b>This is useful to know:</b> When a polynomial is factored like this:</p>
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<p align="center" class="large">f(x) = (x−a)(x−b)(x−c)...</p>
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<p class="larger center">Then a, b, c, etc are the <b>roots</b>!</p>
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<p>So Linear Factors and Roots are related, know one and we can find the other.</p>
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<p>(Read <a href="polynomials-remainder-factor.html">The Factor Theorem</a> for more details.)</p>
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<div class="example">
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<h3>Example: f(x) = (x<sup>3</sup>+2x<sup>2</sup>)(x−3)</h3>
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<p>We see "(x−3)", and that means that 3 is a root (or "zero") of the function.</p>
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<p><b>Sure?</b></p>
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<p>Well, let us put "3" in place of x:</p>
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<p align="center" class="larger">f(x) = (3<sup>3</sup>+2·3<sup>2</sup>)(3−3)</p>
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<p align="center" class="larger">f(3) = (3<sup>3</sup>+2·3<sup>2</sup>)(<span class="hi">0</span>)</p>
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<p><b>Yes!</b> f(3)=0, so 3 is a root.</p>
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</div>
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<h2>How to Check</h2>
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<p>Found a root? <b>Check it! </b></p>
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<p>Simply put the root in place of "x": the polynomial should be equal to zero.</p>
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<div class="example">
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<h3>Example: 2x<sup>3</sup>−x<sup>2</sup>−7x+2</h3>
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<p>The polynomial is degree 3, and could be difficult to solve. So let us plot it first:</p>
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<p align="center"><img src="images/graph-2x3mx2m7xp2.gif" alt="2x^3−x^2−7x+2" width="248" height="134" /></p>
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<p>The curve crosses the x-axis at three points, and one of them <b>might be at 2</b>. We can check easily, just put "2" in place of "x":</p>
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<p align="center"><b>f(2)</b> = 2(2)<sup>3</sup>−(2)<sup>2</sup>−7(2)+2 <br>
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= 16−4−14+2 <br>
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= <b>0</b></p>
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<p>Yes! <b>f(2)=0</b>, so we have found a root!</p>
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<p align="center"> </p>
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<p>How about where it crosses near <b>−1.8</b>:</p>
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<p align="center"><b>f(−1.8)</b> = 2(−1.8)<sup>3</sup>−(−1.8)<sup>2</sup>−7(−1.8)+2 <br>
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= −11.664−3.24+12.6+2 <br>
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= <b>−0.304</b></p>
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<p>No, it isn't equal to zero, so −1.8 will not be a root (but it may be close!)</p>
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</div>
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<p>But we <b>did</b> discover one root, and we can use that to simplify the polynomial, like this</p>
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<div class="example">
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<h3>Example (continued): 2x<sup>3</sup>−x<sup>2</sup>−7x+2 </h3>
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<p>So, <b>f(2)=0</b> is a root ... that means we also know a factor:</p>
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<p align="center"><span class="large">(x−2)</span> must be a factor of <span class="large">2x<sup>3</sup>−x<sup>2</sup>−7x+2</span></p>
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<p> </p>
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<p>Next, divide <span class="large">2x<sup>3</sup>−x<sup>2</sup>−7x+2</span> by <span class="large">(x−2)</span> using <a href="polynomials-division-long.html">Polynomial Long Division</a> to find:</p>
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<p align="center"><span class="large">2x<sup>3</sup>−x<sup>2</sup>−7x+2</span> = <span class="large">(x−2)(2x<sup>2</sup>+3x−1)</span></p>
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<p align="center"> </p>
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<p>So now we can solve <span class="large">2x<sup>2</sup>+3x−1</span> as a Quadratic Equation and we will know all the roots.</p>
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</div>
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<p>That last example showed how useful it is to find just one root. Remember:</p>
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<div class="center80">
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<p>If we find one root, we can then <b>reduce the polynomial by one degree</b> and this may be enough to solve the whole polynomial.</p>
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</div>
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<h2>How Far Left or Right</h2>
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<p>When trying to find roots, <b>how far left and right</b> of zero should we go?</p>
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<p>There is a way to tell,<b></b> and there are a few calculations to do, but it is all simple arithmetic. Read <a href="polynomials-bounds-zeros.html">Bounds on Zeros</a> for all the details.</p>
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<h2>Have We Got All The Roots?</h2>
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<p>There is an easy way to know <b>how many roots</b> there are. The <a href="fundamental-theorem-algebra.html">Fundamental Theorem of Algebra</a> says:</p>
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<div class="center80">
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<p align="center"><span class="larger">A polynomial of degree <b>n</b> ... <br>
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...
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has <b>n</b> roots (zeros)</span><br />
|
||
but we may need to use <a href="../numbers/complex-numbers.html">complex numbers</a></p>
|
||
</div>
|
||
<p>So: <b>number of roots = the degree of polynomial</b>.</p>
|
||
<div class="example">
|
||
<h3>Example: 2x<sup>3</sup> + 3x − 6</h3>
|
||
<p>The degree is 3 (because the largest exponent is 3), and so:</p>
|
||
<p align="center" class="larger">There are <b>3</b> roots.</p>
|
||
</div>
|
||
<h2>But Some Roots May Be Complex</h2>
|
||
<p>Yes, indeed, some roots may be <a href="../numbers/complex-numbers.html">complex numbers</a> (ie have an <a href="../numbers/imaginary-numbers.html">imaginary</a> part), and so will not show up as a simple "crossing of the x-axis" on a graph.</p>
|
||
<p>But there is an interesting fact:</p>
|
||
<p align="center" class="larger">Complex Roots <b>always come in pairs</b>!</p>
|
||
<p align="center"><img src="images/complex-conjugate-pair.gif" alt="Complex Conjugate Pairs" width="227" height="171" />
|
||
</p>
|
||
<p>So we either get <b>no</b> complex roots, or <b>2</b> complex roots, or <b>4</b>, etc... Never an odd number.</p>
|
||
<p>Which means we automatically know this:</p>
|
||
<div class="simple">
|
||
<table border="0" align="center">
|
||
<tr>
|
||
<th>Degree</th>
|
||
<th align="center">Roots</th>
|
||
<th align="center">Possible Combinations</th>
|
||
</tr>
|
||
<tr align="center">
|
||
<td>1</td>
|
||
<td>1</td>
|
||
<td> 1 Real Root </td>
|
||
</tr>
|
||
<tr align="center">
|
||
<td> 2 </td>
|
||
<td>2</td>
|
||
<td>2 Real Roots, <b>or</b> 2 Complex Roots </td>
|
||
</tr>
|
||
<tr align="center">
|
||
<td>3</td>
|
||
<td>3</td>
|
||
<td>3 Real Roots, <b>or</b> 1 Real and 2 Complex Roots</td>
|
||
</tr>
|
||
<tr align="center">
|
||
<td>4</td>
|
||
<td>4</td>
|
||
<td>4 Real Roots, <b>or</b> 2 Real and 2 Complex Roots, <b>or</b> 4 Complex Roots </td>
|
||
</tr>
|
||
<tr align="center">
|
||
<td>etc</td>
|
||
<td> </td>
|
||
<td>etc!</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
<h2>Positive or Negative Roots?</h2>
|
||
<p>There is also a special way to tell how many of the roots are <b>negative</b> or <b>positive</b> called the <a href="polynomials-rule-signs.html">Rule of Signs</a> that you may like to read about.</p>
|
||
<h2>Multiplicity of a Root</h2>
|
||
<p>Sometimes a factor appears more than once. We call that <b>Multiplicity</b>:</p>
|
||
<div class="center80">
|
||
<p align="center"><b>Multiplicity</b> is how often a certain root is part of the factoring.</p>
|
||
</div>
|
||
<div class="example">
|
||
<h3>Example: f(x) = (x−5)<sup>3</sup>(x+7)(x−1)<sup>2</sup></h3>
|
||
<p>This could be written out in a more lengthy way like this:</p>
|
||
<p align="center" class="larger"> f(x) = (x−5)(x−5)(x−5)(x+7)(x−1)(x−1)</p>
|
||
<p><span class="large"> (x−5)</span> is used 3 times, so the root "5" has a multiplicity of <b>3</b>, likewise <span class="large">(x+7)</span> appears once and<span class="large"> (x−1)</span> appears twice. So:</p>
|
||
<ul>
|
||
<li>the root <span class="larger">+5</span> has a multiplicity of <b>3</b></li>
|
||
<li>the root <span class="larger">−7</span> has a multiplicity of <b>1</b> (a "simple" root)</li>
|
||
<li>the root <span class="larger">+1</span> has a multiplicity of <b>2</b></li>
|
||
</ul>
|
||
</div>
|
||
<p class="larger">Q: Why is this useful? <br />
|
||
A: It makes the graph behave in a special way!</p>
|
||
<p>When we see a factor like <b>(x-r)<sup>n</sup></b>, "n" is the multiplicity, and</p>
|
||
<ul>
|
||
<li>even multiplicity <b>just touches the axis</b> at "r" (and otherwise stays one side of the x-axis)</li>
|
||
<li>odd multiplicity <b>crosses the axis</b> at "r" (changes from one side of the x-axis to the other)</li>
|
||
</ul>
|
||
<p>We can see it on this graph:</p>
|
||
<div class="example">
|
||
<h3>Example: f(x) = (x−2)<sup>2</sup>(x−4)<sup>3</sup></h3>
|
||
<p><span class="larger">(x−2)</span> has <b>even multiplicity</b>, so it just touches the axis at x=2</p>
|
||
<p><span class="larger">(x−4)</span> has <b>odd multiplicity</b>, so it crosses the axis at x=4</p>
|
||
<p>Like this:</p>
|
||
<p align="center"><img src="images/polynomial-multiplicity-example.gif" alt="(x−2)^2(x−4)^3" width="320" height="194" /></p>
|
||
</div>
|
||
<h2>Summary</h2>
|
||
<ul>
|
||
<div class="bigul">
|
||
<li>We can directly solve polynomials of Degree 1 (linear) and 2 (quadratic)</li>
|
||
<li>For Degree 3 and up, graphs can be helpful</li>
|
||
<li>It is also helpful to:
|
||
<ul>
|
||
<li>Know how far left or right the roots may be</li>
|
||
<li>Know how many roots (the same as its degree)</li>
|
||
<li>Estimate how many may be complex, positive or negative</li>
|
||
</ul>
|
||
</li>
|
||
<li>Multiplicity is how often a certain root is part of the factoring.</li>
|
||
<br />
|
||
</div>
|
||
</ul>
|
||
<div class="questions">
|
||
<script type="text/javascript">getQ(466,467,468,469,1116,1117, 2276, 2277, 2278, 2279);</script> </div>
|
||
<div class="related"> <a href="introduction.html">Introduction to Algebra</a> <a href="definitions.html">Algebra − Basic Definitions</a> <a href="index.html">Algebra Index</a> </div>
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