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363 lines
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<title>Fundamental Theorem of Algebra</title>
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<h1 align="center">Fundamental Theorem of Algebra</h1>
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<p>The "Fundamental Theorem of Algebra" is <b>not</b> the start of algebra or anything, but it does say something interesting about <a href="polynomials.html">polynomials</a>:</p>
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<div class="center80">
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<p align="center"><span class="larger">Any polynomial of degree <b>n</b> has <b>n</b> roots</span><br />
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but we may need to use complex numbers</p>
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</div>
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<p>Let me explain: </p>
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<div class="dotpoint">
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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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</div>
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<div class="dotpoint">
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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 class="center">... the <a href="../exponent.html">largest exponent</a> of that variable.</p>
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<p class="center"><img src="images/degree-example-a.svg" alt="polynomial" /></p>
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</div>
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<div class="dotpoint">
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<p>A "root" (or "zero") is where the <b>polynomial is equal to zero</b>.</p>
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<p class="center"><img src="images/roots.svg" alt="roots (zeros)" /></p>
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</div> <p>So, a polynomial of degree 3 will have 3 roots (places where the polynomial is equal to zero).
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A polynomial of degree 4 will have 4 roots. And so on.</p>
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<div class="example">
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<h3>Example: what are the roots of <b>x<sup>2</sup> − 9</b>?</h3>
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<p><b>x<sup>2</sup> − 9</b> has a degree of 2 (the largest exponent of x is 2), so there are 2 roots.</p>
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<p>Let us solve it. We want it to be equal to zero: </p>
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<div class="so"><b>x<sup>2</sup> − 9 = 0</b></div>
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<p>Add 9 to both sides:</p>
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<div class="so"><b>x<sup>2</sup> = +9</b></div>
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<p>Then take the square root of both sides:</p>
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<div class="so"><b>x = ±3</b></div>
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<p>So the roots are <b>−3</b> and <b>+3</b></p>
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<p class="center"><img src="images/x2-9.gif" width="178" height="229" alt="x^2 - 9" /></p>
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</div>
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<p>And there is something else of interest:</p>
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<div class="dotpoint">
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<p>A polynomial <b>can be rewritten like this</b>:</p>
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</div>
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<p align="center"><img src="images/polynomial-factoring.svg" alt="Polynomial Factoring" /></p>
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<div class="words"> The factors like <span class="large">(x−r<sub>1</sub>)</span> are called <b>Linear Factors</b>, because they make a <b>line</b> when we plot them. </div>
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<div class="example">
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<h3>Example: <b>x<sup>2</sup> − 9</b></h3>
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<p>The roots are <b>r<sub>1</sub> = −3</b> and <b>r<sub>2</sub> = +3</b> (as we discovered above) so the factors are:</p>
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<div class="so">x<sup>2</sup> − 9 = <b>(x+3)(x−3)</b></div>
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<p>(in this case <b>a</b> is equal to <b>1</b> so I didn't put it in)</p>
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<p align="center" class="larger">The Linear Factors are<b> (x+3)</b> and <b>(x−3)</b></p>
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</div>
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<p>So knowing the <b>roots</b> means we also know the <b>factors</b>. </p>
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<p>Here is another example:</p>
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<div class="example">
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<h3>Example: 3x<sup>2</sup> − 12</h3>
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<p> It is degree 2, so there are 2 roots.</p>
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<p>Let us find the roots: We want it to be equal to zero:</p>
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<div class="so">3x<sup>2</sup> − 12 = 0</div>
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<p>3 and 12 have a common factor of 3:</p>
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<div class="so"> 3(x<sup>2</sup> − 4) = 0</div>
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<p>We can solve <b>x<sup>2</sup> − 4</b> by moving the <b>−4</b> to the right and taking square roots:</p>
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<div class="so">x<sup>2</sup> = 4</div>
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<div class="so">x = ±2 </div>
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<p>So the roots are:</p>
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<p align="center" class="large">x = −2 and x = +2</p>
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<p>And so the factors are:</p>
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<p align="center" class="large">3x<sup>2</sup> − 12 = 3(x+2)(x−2)</p>
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</div>
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<p>Likewise, when we know the <b>factors</b> of a polynomial we also know the <b>roots</b>.</p>
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<div class="example">
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<h3>Example: 3x<sup>2 </sup>− 18x<sup> </sup>+ 24</h3>
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<p>It is degree 2 so there are 2 factors.</p>
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<p align="center"><span class="larger">3x<sup>2 </sup>− 18x<sup> </sup>+ 24 = <span class="large">a(x−r<sub>1</sub>)(x−r<sub>2</sub>)</span></span></p>
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<p>I just happen to know this is the factoring:</p>
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<p align="center" class="larger">3x<sup>2 </sup>− 18x<sup> </sup>+ 24 = <span class="large">3(x−2<sub></sub>)(x−4<sub></sub>)</span></p>
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<p>And so the roots (zeros) are:</p>
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<ul>
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<li>+2</li>
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<li>+4</li>
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</ul>
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<p>Let us check those roots:</p>
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<p align="center">3(2)<sup>2 </sup>− 18(2)<sup> </sup>+ 24 = 12 − 36 + 24 = <b>0</b></p>
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<p align="center">3(4)<sup>2 </sup>− 18(4)<sup> </sup>+ 24 = 48 − 72 + 24 = <b>0</b></p>
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<p>Yes! The polynomial is zero at x = +2 and x = +4</p>
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</div>
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<h2>Complex Numbers</h2>
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<p>We <b>may</b> need to use Complex Numbers to make the polynomial equal to zero.</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>And here is an example:</p>
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<div class="example">
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<h3>Example: x<sup>2</sup>−x+1</h3>
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<p>Can we make it equal to zero?</p>
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<p align="center" class="large">x<sup>2</sup>−x+1 = 0 </p>
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<p>Using the <a href="../quadratic-equation-solver.html">Quadratic Equation Solver</a> the answer (to 3 decimal places) is:</p>
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<div class="example2">
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<table width="400" border="0" align="center">
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<tr align="center">
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<td class="larger">0.5 − 0.866<b>i </b></td>
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<td>and</td>
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<td class="larger">0.5 + 0.866<b>i </b></td>
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</tr>
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</table>
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</div>
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<p>They are complex numbers! But they still work.</p>
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<p>And so the factors are:</p>
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<p align="center" class="larger">x<sup>2</sup>−x+1 = ( x − <span class="large">(0.5−0.866<b>i </b>)</span> )( x − <span class="large">(0.5+0.866<b>i </b>)</span> )</p>
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</div>
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<h2>Complex Pairs</h2>
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<p>So the roots <span class="large">r<sub>1</sub>, r<sub>2</sub>, ... etc</span> may be Real or Complex Numbers.</p>
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<p>But there is something interesting... </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="../numbers/images/complex-conjugate-pair.svg" alt="Complex Conjugate Pairs" /></p>
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<p>You saw that in our example above:</p>
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<div class="example">
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<h3>Example: x<sup>2</sup>−x+1</h3>
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<p>Has these roots:</p>
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<div class="example2">
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<table width="400" border="0" align="center">
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<tr align="center">
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<td class="larger">0.5 − 0.866<b>i </b></td>
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<td>and</td>
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<td class="larger">0.5 + 0.866<b>i </b></td>
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</tr>
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</table>
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</div>
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</div>
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<p>The pair are actually complex conjugates (where we <b>change the sign in the middle</b>) like this:</p>
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<p align="center"><img src="../numbers/images/complex-conjugate.svg" alt="Complex Conjugate" /></p>
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<p>Always in pairs? Yes (unless the polynomial has complex coefficients, but we are only looking at polynomials with real coefficients here!)</p>
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<p>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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<p>And <b>never</b> 1, 3, 5, etc.</p>
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<p>Which means we automatically know this:</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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<th>Degree</th>
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<th align="center">Roots</th>
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<th align="center">Possible Combinations</th>
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</tr>
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<tr align="center">
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<td>1</td>
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<td>1</td>
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<td> 1 Real Root </td>
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</tr>
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<tr align="center">
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<td> 2 </td>
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<td>2</td>
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<td>2 Real Roots, <b>or</b> 2 Complex Roots </td>
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</tr>
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<tr align="center">
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<td>3</td>
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<td>3</td>
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<td>3 Real Roots, <b>or</b> 1 Real and 2 Complex Roots</td>
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</tr>
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<tr align="center">
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<td>4</td>
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<td>4</td>
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<td>4 Real Roots, <b>or</b> 2 Real and 2 Complex Roots, <b>or</b> 4 Complex Roots </td>
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</tr>
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<tr align="center">
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<td>etc</td>
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<td> </td>
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<td>etc!</td>
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</tr>
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</table>
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</div>
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<p>And so:</p>
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<p align="center"> When the degree is odd (1, 3, 5, etc) there is <b>at least one real root</b> ... guaranteed!</p>
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<div class="example">
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<h3>Example: 3x−6</h3>
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<p>The degree is 1. </p>
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<p><b>There is one real root</b> </p>
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<p>At +2 actually:</p>
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<p align="center"><img src="images/graph-3xm6.gif" alt="3x-6" width="136" height="147" />: </p>
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<p>You can actually see that it <b>must go through the x-axis</b> at some point.</p>
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</div>
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<h2>But Real is also Complex!</h2>
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<p>I have been saying "Real" and "Complex", but Complex Numbers do <b>include</b> the Real Numbers.</p>
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<div class="center80">
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<p>So when I say there are <i>"2 Real, and 2 Complex Roots"</i>, I should be saying something like <i>"2 Purely Real (no Imaginary part), and 2 Complex (with a non-zero Imaginary Part) Roots"</i> ...</p>
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<p align="center">... but that is a lot of words that sound confusing ...</p>
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<p align="center">... so I hope you don't mind my (perhaps too) simple language.</p>
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</div>
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<h2>Don't Want Complex Numbers?</h2>
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<p>If we <b>don't</b> want Complex Numbers, we can multiply pairs of complex roots together:</p>
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<div class="def">
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<p align="center"><span class="larger">(a + b<b>i</b>)(a − b<b>i</b>) = a<sup>2</sup></span><span class="larger"> + b<sup>2</sup></span></p>
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</div>
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<p>We get a <a href="quadratic-equation.html">Quadratic Equation</a> with no Complex Numbers ... it is purely Real.</p>
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<div class="words">
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<p>That type of Quadratic (where we can't "reduce" it any further without using Complex Numbers) is called an <b>Irreducible Quadratic</b>.</p>
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</div>
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<div class="words">
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<p>And remember that simple factors like <span class="large">(x-r<sub>1</sub>)</span> are called <b>Linear Factors</b></p>
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</div>
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<p> </p>
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<div class="center80">
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<p>So a polynomial can be factored into all Real values using:</p>
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<ul>
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<li><b>Linear Factors</b>, and </li>
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<li><b>Irreducible Quadratics</b></li>
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</ul>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: x<sup>3</sup>−1</h3>
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<p align="center" class="larger">x<sup>3</sup>−1 = (x−1)(x<sup>2</sup>+x+1)</p>
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<p>It has been factored into:</p>
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<ul>
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<li>1 linear factor: <span class="larger">(x−1)</span></li>
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<li>1 irreducible quadratic factor: <span class="larger">(x<sup>2</sup>+x+1)</span></li>
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</ul>
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<p>To factor <span class="larger">(x<sup>2</sup>+x+1)</span> further we need to use Complex Numbers, so it is an "Irreducible Quadratic"</p>
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</div>
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<h2>How do we know if the Quadratic is Irreducible?</h2>
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<p>Just calculate the "discriminant": <span class="time">b<sup>2</sup> - 4ac</span></p>
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<p>(Read <a href="quadratic-equation.html">Quadratic Equations</a> to learn more about the discriminant.)</p>
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<p align="center" class="larger">When <b>b<sup>2</sup> − 4ac</b> is negative, the Quadratic has Complex solutions, <br />
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and so is "Irreducible" </p>
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<div class="example">
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<h3>Example: 2x<sup>2</sup>+3x+5</h3>
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<p>a = 2, b = 3, and c = 5:</p>
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<p align="center"><b>b<sup>2</sup> − 4ac</b> = 3<sup>2</sup> − 4×2×5 = 9−40 = <b>−31</b></p>
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<p>The discriminant is negative, so it is an "Irreducible Quadratic"</p>
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</div>
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<p> </p>
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<h2>Multiplicity</h2>
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<p>Sometimes a factor appears more than once. That is its <b>Multiplicity</b>.</p>
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<div class="example">
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<h3>Example: x<sup>2</sup>−6x+9</h3>
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<p align="center" class="larger">x<sup>2</sup>−6x+9 = (x−3)(x−3)</p>
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<p>"(x−3)" appears twice, so the root "3" has <b>Multiplicity of 2</b></p>
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</div>
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<p>The <b>Multiplicities</b> are included when we say "a polynomial of degree <b>n</b> has <b>n</b> roots".</p>
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<div class="example">
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<h3>Example: x<sup>4</sup>+x<sup>3</sup></h3>
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<p>There <b>should be</b> 4 roots (and 4 factors), right?</p>
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<p>Factoring is easy, just factor out <span class="larger">x<sup>3</sup></span>:</p>
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<p align="center" class="large">x<sup>4</sup>+x<sup>3</sup> = x<sup>3</sup>(x+1) = x·x·x·(x+1)</p>
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<p align="center">there are 4 factors, with "x" appearing 3 times.</p>
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<p>But there seem to be only 2 roots, at <b>x=−1</b> and <b>x=0</b>:</p>
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<p align="center"><img src="images/graph-x4px3.gif" alt="x^4+x^3" width="302" height="133" /></p>
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|
<p>But counting Multiplicities there are actually 4:</p>
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|
<ul>
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|
<li>"x" appears three times, so the root "0" has a <b>Multiplicity of 3</b></li>
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|
<li>"x+1" appears once, so the root "−1" has a <b>Multiplicity of 1</b></li>
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|
</ul>
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|
<p align="center" class="larger">Total = 3+1 = 4</p>
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</div>
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<h2>Summary</h2>
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<ul>
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<div class="bigul">
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<li>A polynomial of degree <b>n</b> has <b>n</b> roots (where the polynomial is zero)</li>
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<li>A polynomial can be <b></b> factored like: <span class="beach"><i><b>a(x−r<sub>1</sub>)(x−r<sub>2</sub>)</b></i></span><i><b>... </b></i>where r<sub>1</sub>, etc are the roots</li>
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<li>Roots may need to be <b>Complex Numbers</b></li>
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<li>Complex Roots <b>always come in pairs</b></li>
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<li>Multiplying a Complex pair gives an <b>Irreducible Quadratic</b></li>
|
|
<li>So a polynomial can be factored into all real factors which are either:
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<ul>
|
|
<li><b>Linear Factors</b> or</li>
|
|
<li><b>Irreducible Quadratics</b></li>
|
|
</ul>
|
|
</li>
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|
<li>Sometimes a factor appears more than once. That is its <b>Multiplicity</b>.</li>
|
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</div>
|
|
</ul>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(478, 479, 480, 481, 1122, 1123, 2252, 2253, 2254, 4013);</script>
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</div>
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<div class="related"><a href="index.html">Algebra Index</a></div>
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