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372 lines
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<h1 align="center">Rational Expressions</h1>
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<p align="center">An expression that 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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<p>It is just like a fraction, but with polynomials.</p>
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<h2>Other Examples:</h2>
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<table border="0" align="center">
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<tr align="center">
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<td width="250"><span class="intbl large"><em>x<sup>3</sup> + 2x − 1</em><strong>6x<sup>2</sup></strong></span></td>
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<td width="250"><span class="intbl large"><em>2x + 9</em><strong>x<sup>4</sup> − x<sup>2</sup></strong></span></td>
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</tr>
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</table>
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<h2>Also</h2>
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<table border="0" align="center">
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<tr align="center">
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<td width="150"><span class="intbl large"><em>1</em><strong>2 − x<sup>2</sup></strong></span></td>
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<td>The top polynomial is "1" which is fine.</td>
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</tr>
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<tr align="center">
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<td> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td><span class="large">2x<sup>2</sup> + 3</span></td>
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<td>Yes it is! As it could also be written:<br><span class="intbl large"><em>2x<sup>2</sup> + 3</em><strong>1</strong></span>
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<br /></td>
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</tr>
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</table>
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<h2>But Not</h2>
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<table border="0" align="center">
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<tr align="center">
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<td width="70"><img src="../images/style/no.svg" alt="not" height="46" /></td>
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<td width="180"><span class="intbl large"><em>2 − √(x)</em><strong>4 − x</strong></span></td>
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<td width="250">the top is not a polynomial (a square root of a variable is not allowed)</td>
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</tr>
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<tr align="center">
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr align="center">
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<td><img src="../images/style/no.svg" alt="not" height="46" /></td>
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<td><img src="images/rational-expression-not-2.gif" alt="NOT" width="55" height="53" /></td>
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<td>1/x is not allowed in a polynomial</td>
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</tr>
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</table>
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<h2>In General</h2>
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<p>A rational function is the ratio of two polynomials P(x) and Q(x) like this</p>
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<p align="center"><span class="large">f(x) = <span class="intbl"><em>P(x)</em><strong>Q(x)</strong></span></span></p>
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<p>Except that Q(x) cannot be zero (and anywhere that <b>Q(x)=0</b> is undefined)</p>
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<h2>Finding Roots of Rational Expressions</h2>
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<table border="0" align="center">
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<tr>
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<td align="center">
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<p>A "root" (or "zero") is where the expression <b>is equal to zero</b>:
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<br /> </p>
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</td>
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</tr>
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<tr>
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<td align="center"><img src="images/inequality-graph-function.svg" alt="Graph of Inequality" /></td>
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</tr>
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</table>
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<p>To find the roots of a <b>Rational Expression</b> we only need to find the the roots of the <b>top polynomial</b>, so long as the Rational Expression is in "Lowest Terms".</p>
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<p>So what does "Lowest Terms" mean?</p>
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<h2>Lowest Terms</h2>
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<p>Well, a <b>fraction</b> is in Lowest Terms when the top and bottom have no common factors.</p>
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<div class="example">
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<h3>Example: Fractions</h3>
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<p class="center"><span class="intbl large">
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<em>2</em>
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<strong>6</strong>
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</span> is <b>not</b> in lowest terms,
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<br> as 2 and 6 have the common factor "2" </p>
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<p>But:</p>
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<p class="center"><span class="intbl large">
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<em>1</em>
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<strong>3</strong>
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</span> <b>is</b> in lowest terms,
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<br> as 1 and 3 have no common factors </p>
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</div>
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<p>Likewise a <b>Rational Expression</b> is in Lowest Terms when the top and bottom have no common factors.</p>
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<div class="example">
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<h3>Example: Rational Expressions</h3>
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<p class="center"><span class="intbl large"><em>x<sup>3</sup>+3x<sup>2</sup></em><strong>2x</strong></span> is <b>not</b> in lowest terms,
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<br> as <span class="large">x<sup>3</sup>+3x<sup>2</sup></span> and <span class="large">2x
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</span> have the common factor "x"</p>
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<p>But</p>
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<p class="center"><span class="intbl large"><em>x<sup>2</sup>+3x</em><strong>2</strong></span> <b>is</b> in lowest terms,
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<br> as <span class="large">x<sup>2</sup>+3x</span> and <span class="large">2</span> have no common factors</p>
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</div>
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<p>So, to find the roots of a <b>rational expression:</b></p>
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<div class="bigul">
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<ul>
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<li>Reduce the rational expression to Lowest Terms,</li>
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<li>Then find the roots of the <b>top polynomial</b></li>
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</ul>
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</div>
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<p>How do we find roots? Read <a href="polynomials-solving.html">Solving Polynomials</a> to learn how.</p>
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<h2>Proper vs Improper</h2>
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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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<td align="center" class="larger">Fractions can be <a href="../proper-fractions.html">proper</a> or <a href="../improper-fractions.html">improper</a>:</td>
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</tr>
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<tr>
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<td><img src="../numbers/images/fraction-types.svg" alt="Fraction Types" /></td>
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</tr>
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<tr>
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<td>(There is nothing wrong with "Improper", it is just a different type)</td>
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</tr>
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</table>
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</div>
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<p>And likewise:</p>
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<p align="center" class="larger"> A Rational Expression can also be <b>proper</b> or <b>improper</b>!</p>
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<p>But what makes a polynomial larger or smaller? </p>
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<p align="center" class="larger">The <a href="degree-expression.html">Degree</a> !</p>
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<div class="def">
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<p>For a polynomial with one variable, the Degree is the largest <a href="../exponent.html">exponent</a> of that variable.</p>
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<h3>Examples of Degree:</h3>
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<table border="0" align="center">
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<tr>
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<td align="center" class="larger"><b>4x</b></td>
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<td width="30"> </td>
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<td>The Degree is <b>1</b> (a variable without an exponent
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<br> actually has an exponent of 1)</td>
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</tr>
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<tr>
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<td align="center" class="larger"> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td align="center" class="larger"><b>4x<sup>3</sup> − x + 3</b></td>
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<td> </td>
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<td>The Degree is <b>3</b> (largest exponent of x)</td>
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</tr>
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</table>
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<br /> </div>
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<br />
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<p>So this is how to know if a rational expression is <b>proper</b> or <b>improper</b>:</p>
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<div class="dotpoint">
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<p><span class="large">Proper</span>: the degree of the top is less than the degree of the bottom.</p>
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<table border="0" align="center">
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<tr align="center">
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<td width="90">Proper:</td>
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<td width="90" nowrap><span class="intbl large"><em>1</em><strong>x + 1</strong></span></td>
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<td width="40"> </td>
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<td width="200">deg(top) <span class="large"><</span> deg(bottom)</td>
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</tr>
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</table>
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<p class="center">Another Example: <span class="intbl large"><em>x</em><strong>x<sup>3</sup> − 1</strong></span></p>
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</div>
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<div class="dotpoint">
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<p><span class="large">Improper</span>: the degree of the top is greater than, or equal to, the degree of the bottom.</p>
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<table border="0" align="center">
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<tr align="center">
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<td width="90">Improper:</td>
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<td width="90" nowrap><span class="intbl large"><em>x<sup>2</sup> − 1</em><strong>x + 1</strong></span></td>
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<td width="40"> </td>
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<td width="200">deg(top) <span class="large">≥</span> deg(bottom)</td>
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</tr>
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</table>
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<p class="center">Another Example: <span class="intbl large"><em>4x<sup>3</sup> − 3</em><strong>5x<sup>3</sup> + 1</strong></span></p>
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<p> </p>
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</div>
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<p>If the polynomial is improper, we can simplify it with <a href="polynomials-division-long.html">Polynomial Long Division</a></p>
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<h2>Asymptotes</h2>
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<p>Rational expressions can have <a href="asymptote.html">asymptotes</a> (a <b>line</b> that a curve approaches as it heads towards infinity)<span class="larger">:</span></p>
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<div class="example">
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<h3>Example: (x<sup>2</sup>-3x)/(2x-2)</h3>
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<table border="0" align="center">
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<tr>
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<td>
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<p>The <a href="../data/function-grapher090b.html?func1=(x^2-3x)/(2x-2)&func2=x/2-1&xmin=-10&xmax=10&ymin=-6.17&ymax=7.17">graph of (x<sup>2</sup>-3x)/(2x-2)</a> has:</p>
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<ul>
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<li>A vertical asymptote at <b>x=1</b></li>
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<li>An oblique asymptote: <b>y=x/2-1</b></li>
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</ul>
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</td>
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<td> </td>
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<td><img src="images/asymptote-example.gif" alt=" Asymptote Example" width="191" height="193" /></td>
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</tr>
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</table>
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</div>
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<p>A rational expression can have:</p>
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<ul>
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<li>any number of vertical asymptotes,</li>
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<li>only zero or one horizontal asymptote,</li>
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<li>only zero or one oblique (slanted) asymptote</li>
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</ul>
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<h2>Finding Horizontal or Oblique Asymptotes</h2>
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<p>It is fairly easy to find them ...</p>
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<p align="center" class="larger">... but it depends on <b>the degree </b>of the<b> top vs bottom </b>polynomial.</p>
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<p align="center"> The one with the larger degree will grow fastest. </p>
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<p>Just like "Proper" and "Improper", but in fact there are <b>four possible cases, </b>shown below.</p>
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<p> </p>
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<p class="center"><img src="images/rational-asymptote-degree.svg" alt="rational asymptote degrees" />
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<br>
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<i>(I show a test value of <b>x=1000</b> for each case, just to show what happens)</i></p>
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<p> </p>
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<p>Let's look at each of those examples in turn:</p>
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<h3>Degree of Top <b>Less </b>Than Bottom</h3>
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<p>The bottom polynomial will dominate, and there is a Horizontal Asymptote at zero.</p>
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<div class="example">
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<h3>Example: f(x) = (3x+1)/(4x<sup>2</sup>+1)</h3>
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<p>When x is 1000:</p>
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<p align="center"><b>f(1000)</b> = 3001/4000001 = <b>0.00075...</b></p>
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<p>And as x gets larger, f(x) gets closer to 0</p>
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</div>
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<p> </p>
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<h3>Degree of Top is <b>Equal To</b> Bottom</h3>
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<p>Neither dominates ... the asymptote is set by the leading terms of each polynomial.</p>
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<div class="example">
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<h3>Example: f(x) = (3x+1)/(4x+1)</h3>
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<p>When x is 1000:</p>
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<p align="center"><b>f(1000)</b> = 3001/4001 = <b>0.750...</b></p>
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<p>And as x gets larger, f(x) gets closer to 3/4</p>
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<p>Why 3/4? Because "3" and "4" are the "leading coefficients" of each polynomial</p>
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</div>
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<p align="center"><img src="images/polynomial-coefficients.svg" alt="polynomial coefficients" />
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<br /> The terms are in order from highest to lowest exponent</p>
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<p>(Technically the 7 is a constant, but here it is easier to think of them all as coefficients.)</p>
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<p>The method is easy:</p>
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<div class="def">
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<p>Divide the leading coefficient of the top polynomial by the leading coefficient of the bottom polynomial.</p>
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</div>
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<p>Here is another example: </p>
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<div class="example">
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<h3>Example: f(x) = (8x<sup>3</sup> + 2x<sup>2</sup> - 5x + 1)/(2x<sup>3</sup> + 15x + 2)</h3>
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<p>The degrees are equal (both have a degree of 3)</p>
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<p>Just look at the leading coefficients of each polynomial:</p>
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<ul>
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<li>Top is <b>8</b> (from 8x<sup>3</sup>)</li>
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<li>Bottom is <b>2</b> (from 2x<sup>3</sup>)</li>
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</ul>
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<p>So there is a Horizontal Asymptote at 8/2 = <b>4</b></p>
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</div>
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<p> </p>
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<h3>Degree of Top is <b>1 Greater </b>Than Bottom</h3>
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<p>This is a special case: there is an <b>oblique asymptote</b>, and we need to find the equation of the line.</p>
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<p>To work it out use <a href="polynomials-division-long.html">polynomial long division</a>: divide the top by the bottom to find the quotient (ignore the remainder).</p>
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<div class="example">
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<h3>Example: f(x) = (3x<sup>2</sup>+1)/(4x+1)</h3>
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<p>The degree of the top is 2, and the degree of the bottom is 1, so there will ne an oblique asymptote</p>
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<p>We need to divide <b>3x<sup>2</sup>+1</b> by <b>4x+1</b> using polynomial long division:</p>
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<p align="center"><img src="images/polynomial-long-division-fraction.gif" alt="polynomial long division" width="176" height="211" /></p>
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<p>The answer is <b>(3/4)x-(3/16)</b> (ignoring the remainder):</p>
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<p align="center" class="larger">Asymptote "equation of line" is: (3/4)x-(3/16)</p>
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</div>
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<p> </p>
|
|
<h3>Degree of Top is <b>More Than 1 Greater </b>Than Bottom</h3>
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<p>When the top polynomial is <b>more than 1 degree</b> higher than the bottom polynomial, there is <b>no horizontal or oblique asymptote</b>.</p>
|
|
<div class="example">
|
|
<h3>Example: f(x) = (3x<sup>3</sup>+1)/(4x+1)</h3>
|
|
<p>The degree of the top is 3, and the degree of the bottom is 1.</p>
|
|
<p>The top is more than 1 degree higher than the bottom so there is <b>no horizontal or oblique asymptote</b>.</p>
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</div>
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<h2>Finding Vertical Asymptotes</h2>
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<p>There is another type of asymptote, which is caused by the <b>bottom polynomial only</b>.</p>
|
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<p align="center" class="larger">But First: make sure the rational expression is in lowest terms!</p>
|
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/asymptote-vertical.svg" alt="Vertical Asymptote" /></p>
|
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<p> </p>
|
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<p>Whenever the <b>bottom polynomial is equal to zero</b> (any of its roots) we get a vertical asymptote.</p>
|
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<p>Read <a href="polynomials-solving.html">Solving polynomials</a> to learn how to find the roots</p>
|
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<p> </p>
|
|
<p>From our example above:</p>
|
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<div style="clear:both"></div>
|
|
<div class="example">
|
|
<h3>Example: (x<sup>2</sup>-3x)/(2x-2)</h3>
|
|
<table border="0" align="center">
|
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<tr>
|
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<td>
|
|
<p>The bottom polynomial is <b>2x-2</b>, which factors into:</p>
|
|
<p align="center" class="large">2(x-1)</p>
|
|
<p>And the factor <b>(x-1)</b> means there is a vertical asymptote at <b>x=1</b> (because 1-1=0)</p>
|
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</td>
|
|
<td> </td>
|
|
<td><img src="images/asymptote-example.gif" alt=" Asymptote Example" width="191" height="193" /></td>
|
|
</tr>
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</table>
|
|
</div>
|
|
<h2>A Full Example </h2>
|
|
<div class="example">
|
|
<h3>Example: Sketch (x−1)/(x<sup>2</sup>−9)</h3>
|
|
<p>First of all, we can factor the bottom polynomial (it is the difference of two squares):</p>
|
|
<p class="center large"><span class="intbl"><em>x−1</em><strong>(x+3)(x−3)</strong></span></p>
|
|
<p>Now we can see:</p>
|
|
<p>The roots of the top polynomial are: <b>+1</b> (this is where it <b>crosses the x-axis</b>)</p>
|
|
<p>The roots of the bottom polynomial are: <b>−3</b> and <b>+3</b> (these are <b>Vertical Asymptotes</b>)</p>
|
|
<p>It <b>crosses the y-axis</b> when x=0, so let us set x to 0:</p>
|
|
<p class="center large">Crosses y-axis at: <span class="intbl">
|
|
<em>0−1</em><strong>(0+3)(0−3)</strong></span> = <span class="intbl"><em>−1</em><strong>−9</strong></span> = <span class="intbl"><em>1</em><strong>9</strong></span></p>
|
|
<p>We also know that the degree of the top is less than the degree of the bottom, so there is a <b>Horizontal Asymptote at 0</b></p>
|
|
<p>So we can sketch all of that information:</p>
|
|
<p align="center"><img src="images/asymptote-example2a.svg" alt=" Sketch of Asymptotes" /></p>
|
|
<p>And now we can sketch in the curve:</p>
|
|
<p align="center"><img src="images/asymptote-example2b.svg" alt=" Sketch of (x-1)/(x^2- 9)" /></p>
|
|
<p>(Compare that to the <a href="../data/function-grapher82ab.html?func1=(x-1)/(x^2-9)&xmin=-6&xmax=6&ymin=-4&ymax=4">plot of (x-1)/(x<sup>2</sup>-9)</a>)</p>
|
|
</div>
|
|
<p> </p>
|
|
<div class="questions">
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<script type="text/javascript">
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getQ(470, 471, 472, 2270, 473, 2271, 1118, 2272, 1119, 2273);
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</script> </div>
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<div class="related"> <a href="rational-expression-operations.html">Using Rational Expressions</a> <a href="polynomials.html">Polynomials</a> <a href="index.html">Algebra Index</a></div>
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