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146 lines
7.3 KiB
HTML
146 lines
7.3 KiB
HTML
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<title>Zero Product Property</title>
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<h1 align="center">Zero Product Property</h1>
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<p>The "Zero Product Property" says that:</p>
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<div class="def">
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<p class="center">If <span class="largest">a × b = 0</span> then <span class="largest">a = 0</span> or <span class="largest">b = 0</span> <br>
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(or both a=0 and b=0)</p>
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</div>
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<p>It can help us solve equations: </p>
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<div class="example">
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<h3>Example: Solve (x−5)(x−3) = 0</h3>
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<p>The "Zero Product Property" says:</p>
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<p align="center">If <span class="large">(x−5)(x−3) = 0</span> then <span class="large">(x−5) = 0</span> <b>or</b> <span class="large">(x−3) = 0</span></p>
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<p>Now we just solve each of those:</p>
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<p align="center">For <span class="large">(x−5) = 0</span> we get <span class="large">x = 5</span></p>
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<p align="center" class="larger"></p>
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<p align="center">For <span class="large">(x−3) = 0</span> we get <span class="large">x = 3</span></p>
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<p align="center" class="larger"></p>
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<p>And the solutions are: </p>
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<p align="center" class="larger">x = <b>5</b>, or x = <b>3</b></p>
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<p>Here it is on a graph:</p>
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<p class="center"><img src="images/x-5-x-3-graph.gif" width="243" height="191" alt="(x-5)(x-3) = 0 graph" /><br>
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<b>y=0 when x=3 or x=5 </b></p>
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</div>
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<h2>Standard Form of an Equation</h2>
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<p>Sometimes we can solve an equation by putting it into Standard Form and then using the Zero Product Property:</p>
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<div class="def">
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<p>The "Standard Form" of an equation is:</p>
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<p align="center"><i>(some expression)</i> <span class="large">= 0</span></p>
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</div>
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<p>In other words, "= 0" is on the right, and everything else is on the left.</p>
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<div class="example">
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<h3>Example: Put x<sup>2</sup> = 7 into Standard Form</h3>
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<p>Answer:</p>
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<p align="center"><span class="large">x<sup>2</sup> − 7 = 0</span><br />
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</p>
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</div>
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<h2>Standard Form and the Zero Product Property</h2>
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<p>So let's try it out:</p>
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<div class="example">
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<h3>Example: Solve 5(x+3) = 5x(x+3)</h3>
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<p>It is tempting to divide by (x+3), but that is <a href="../numbers/dividing-by-zero.html">dividing by zero</a> when <span class="larger">x = −3</span></p>
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<p>So instead we can use "Standard Form":</p>
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<p align="center" class="larger">5(x+3) − 5x(x+3) = 0</p>
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<p>Which can be simplified to:</p>
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<p align="center" class="larger">(5−5x)(x+3) = 0</p>
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<p align="center" class="larger">5(1−x)(x+3) = 0</p>
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<p>Then the "Zero Product Property" says:</p>
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<p align="center" class="larger">(1−x) = 0, or (x+3) = 0</p>
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<p>And the solutions are: </p>
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<p align="center" class="larger">x = <b>1</b>, or x = <b>−3</b></p>
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</div>
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<p>And another example:</p>
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<div class="example">
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<h3>Example: Solve <b>x<sup>3</sup> = 25x</b></h3>
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<p>It is tempting to divide by x, but that is <a href="../numbers/dividing-by-zero.html">dividing by zero</a> when <span class="larger">x = 0</span></p>
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<p>So let's use Standard Form and the Zero Product Property.</p>
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<p> </p>
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<p>Bring all to the left hand side:</p>
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<p align="center" class="larger">x<sup>3</sup> − 25x = 0</p>
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<p>Factor out x:</p>
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<p align="center" class="larger"></p>
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<p align="center" class="larger">x(x<sup>2</sup> − 25) = 0</p>
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<p><span class="larger">x<sup>2</sup> − 25</span> is a <a href="special-binomial-products.html">difference of squares</a>, and can be factored into <span class="larger">(x − 5)(x + 5)</span>:</p>
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<p align="center" class="larger">x(x − 5)(x + 5) = 0</p>
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<p>Now we can see three possible ways it could end up as zero:</p>
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<p align="center" class="larger">x = <b>0</b>, or x = <b>5</b>, or x = <b>−5</b></p>
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</div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(509, 510, 1140, 4030, 1141, 4031, 2292, 4032, 2293, 4033);</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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