SECTION 11.5 Partial Fraction Decomposition 813 Identifying Proper and Improper Rational Expressions Determine whether the rational expression is proper or improper. If the expression is improper, rewrite it as the sum of a polynomial and a proper rational expression. (a) + + + x x x 5 3 2 2 (b) − + − x x x 6 5 3 2 2 Solution EXAMPLE 1 (a) The numerator, +x 5, is a polynomial of degree 1, and the denominator, + + x x3 2, 2 is a polynomial of degree 2. Since the degree of the numerator is less than the degree of the denominator, the rational expression is proper. (b) The numerator, − + x x 6 5, 2 is a polynomial of degree 2, and the denominator, −x3 2, is a polynomial of degree 1. Since the degree of the numerator is greater than the degree of the denominator, the rational expression is improper. We use long division to rewrite this expression as the sum of a polynomial and a proper rational expression: ) + − − + − + − x x x x x x x x 2 1 3 2 6 5 6 4 3 5 3 2 7 2 2 So, − + − = + + − x x x x x 6 5 3 2 2 1 7 3 2 . 2 The reverse procedure, starting with the rational expression − + − x x x 5 1 12 2 and writing it as the sum (or difference) of the two simpler fractions +x 3 4 and −x 2 3 , is referred to as partial fraction decomposition, and the two simpler fractions are called partial fractions. Decomposing a rational expression into a sum of partial fractions is important in solving certain types of calculus problems. This section presents a systematic way to decompose rational expressions. Recall that a rational expression is the ratio of two polynomials, say P and ≠ Q 0. Recall also that a rational expression P Q is called proper if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. Otherwise, the rational expression is called improper. Also, we assume P and Q are in lowest terms. That is, P and Q have no common factors. Now Work PROBLEMS 5 AND 13 By using long division, every improper rational expression can be written as the sum of a polynomial and a proper rational expression. Because of this, we restrict the following discussion to proper rational expressions. The partial fraction decomposition of the rational expression P Q , in lowest terms, depends on the factors of the denominator Q. Recall from Section 4.3 that any polynomial with real coefficients can be factored over the real numbers into a product of linear and/or irreducible quadratic factors.
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