5.4 - Partial Fractions

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<?xml version="1.0" encoding="iso-8859-1"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>5.4 - Partial Fractions</title> <meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" /> <link href=". ./m116.css" rel="stylesheet" type="text/css" /> </head> <body> <h1>5.4 - Partial Fractions</h1> <h2>Addition of Rational Expressions</h2> <p>In arithmetic, you learned how to add fractions. You found the least common denominator, and then multiplied both the numerator and denominator of each term by what was needed to complete the common denominator. </p> <p><img src="partial1.gif" alt="1/(x-1) + (2x-1)/(x^2+3) = (3x^2-3x+4)/((x-1) (x^2+3))" width="378" height="180" class="imgrt" />In algebra, you have carried that process on to addition of rational expressions. You once again multiplied the numerator and denominator of each term by what was missing from the denominator of that term.</p> <p>With Partial Fraction Decomposition, we're going to reverse the process and decompose a rational expression into two or more simpler proper rational expressions that were added together.</p> <h2>Partial Fraction Decomposition</h2> <p>Partial Fraction Decomposition only works for proper rational expressions, that is, the degree of the numerator must be less than the degree of the denominator. If it is not, then you must perform long division first, and then perform the partial fraction decomposition on the rational part (the remainder over the divisor). After you've done the partial fraction decomposition, just add back in the quotient part from the long division. </p> <p>When discussing polynomials in <a href=". ./polynomials/theorem.html">section 3.4</a>, we learned that every polynomial with real coefficients can be factored using only linear and irreducible quadratic factors. This means that there are only two types of factors that we have to worry about.</p> <h3>Linear Factors</h3> <p>If the partial fractions we're decomposing the rational expression into must be proper, then the only thing that can be over a linear factor is a constant. So, for every linear factor in the denominator, you will need a constant in over that in the numerator.</p> <h3>Irreducible Quadratic factors.</h3> <p>If the partial fractions we're decomposing the rational expression into must be proper, then an irreducible quadratic factor could have a linear term and/or a constant term in the numerator. So,
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5.4 - Partial Fractions - &lt;?xml version=&quot;1.0&quot;...

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