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168 lines
9.4 KiB
HTML
168 lines
9.4 KiB
HTML
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<title>Using Rational Expressions</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" --> <h1 align="center">Using Rational Expressions</h1>
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<p>A <a href="rational-expression.html">Rational Expression</a> is the ratio of two <a href="polynomials.html">polynomials</a>:</p>
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<p align="center"><img src="images/rational-expression.svg" alt="Rational Expression" /></p>
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<h2>Using Rational Expressions</h2>
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<p>Using Rational Expressions is very similar to <a href="rational-numbers-operations.html">Using Rational Numbers</a> (you may like to read that first).</p>
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<h2>Adding Rational Expressions</h2>
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<p>The easiest way to add Rational Expressions is to use the <a href="../numbers/common-denominator.html">common denominator</a> method:</p>
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<div class="beach">
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<table border="0" align="center">
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<tr>
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<td><img src="../numbers/images/common-factor-formula.gif" alt="common-factor-formula" width="315" height="60" /></td>
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</tr>
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</table>
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</div>
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<p>Like in this example:</p>
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<div class="example">
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<h3>Example: </h3><p class="center large"><span class="intbl"><em>2</em><strong>x−2</strong></span> + <span class="intbl"><em>3</em><strong>x+1</strong></span> = <span class="intbl"><em>2 × (x+1) + (x−2) × 3</em><strong>(x−2) × (x+1)</strong></span></p>
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<div class="example2"></div>
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<p>And then simplify to:</p>
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<p class="center large">= <span class="intbl">
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<em>2x+2 + 3x−6</em>
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<strong>x<sup>2</sup>+x−2x−2</strong>
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</span></p>
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<p class="center large">= <span class="intbl">
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<em>5x−4</em>
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<strong>x<sup>2</sup>−x−2</strong>
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</span></p>
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</div>
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<h2>Subtracting Rational Expressions</h2>
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<p>Subtracting is just like Adding:</p>
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<div class="example">
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<h3>Example: </h3>
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<p class="center large"><span class="intbl">
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<em>2</em>
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<strong>x−2</strong>
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</span> − <span class="intbl">
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<em>3</em>
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<strong>x+1</strong>
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</span> = <span class="intbl">
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<em>2 × (x+1) − (x−2) × 3</em>
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<strong>(x−2) × (x+1)</strong>
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</span></p>
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<div class="example2"></div>
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<p>And then simplify to:</p>
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<p class="center large">= <span class="intbl">
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<em>2x+2 − (3x−6)</em>
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<strong>x<sup>2</sup>+x−2x−2</strong>
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</span></p>
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<p class="center large">= <span class="intbl">
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<em>−x + 8</em>
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<strong>x<sup>2</sup>−x−2</strong>
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</span></p>
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</div>
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<h2>Multiplication</h2>
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<p>To multiply two Rational Expressions, just multiply the tops and bottoms separately, like this:</p>
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<div class="example">
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<h3>Example: </h3>
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<p class="center large"><span class="intbl"><em>2</em><strong>x−2</strong></span> × <span class="intbl"><em>3</em><strong>x+1</strong></span> = <span class="intbl"><em>2×3</em><strong>(x−2)(x+1)</strong></span></p>
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<p>And then simplify to:</p>
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<p class="center large"> = <span class="intbl"><em>6</em><strong>x<sup>2</sup>−x−2</strong></span></p>
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</div>
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<h2>Division</h2>
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<p>To divide two Rational Expressions, first flip the second expression over (make it a reciprocal) and then do a multiply like above:</p>
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<div class="example">
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<h3>Example: </h3>
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<p>First flip the second one over and make it a multiply:</p><p class="center large"><span class="intbl"><em>2</em><strong>x<span class="large">−</span>2</strong>
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</span> / <span class="intbl">
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<em>3</em><strong>x+1</strong></span><span class="large"> = </span><span class="intbl"><em>2</em><strong>x<span class="large">−</span>2</strong></span><span class="large"> × </span><span class="intbl"><em>x+1</em><strong>3</strong></span></p>
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<p>Then do the multiply:</p>
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<p class="center large"><span class="intbl">
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<em>2</em>
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<strong>x<span class="large">−</span>2</strong>
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</span><span class="large"> × </span><span class="intbl"><em>x+1</em><strong>3</strong></span> = <span class="intbl"><em>2(x+1)</em><strong>3(x<span class="large">−</span>2)</strong></span></p>
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</div>
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<h2>Simplifying</h2>
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<p>When simplifying a rational function be careful to respect where the lower polynomial is equal to zero</p>
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<div class="example">
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<h3>Example: </h3>
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<p class="center large"><span class="intbl"><em>x<sup>2</sup>−1</em><strong>x+1</strong></span> is undefined when x=−1</p>
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<p align="center">Its <a href="../sets/domain-range-codomain.html">Domain</a> (the values that can go into the expression) does not include −1</p>
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<p> </p>
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<p>Now, we can factor <span class="larger">x<sup>2</sup>−1</span> into <span class="larger">(x−1)(x+1)</span> so we get:</p>
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<p class="center large"><span class="intbl"><em>(x−1)(x+1)</em><strong>(x+1)</strong></span></p>
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<p>It is now tempting to cancel <span class="large">(x+1)</span> from top and bottom to produce:</p>
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<p align="center"> <span class="large">x − 1</span></p>
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<p align="center">Its Domain now <b>does</b> include −1</p>
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<p>But it is now a different function because it has a different Domain.</p>
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</div>
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<p> </p>
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<div class="related"> <a href="polynomials.html">Polynomials</a> <a href="index.html">Algebra Index</a></div><!-- #EndEditable --></div>
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