814 CHAPTER 11 Systems of Equations and Inequalities This means that the denominator Q of the rational expression P Q contains only factors of one or both of the following types: • Linear factors of the form −x a, where a is a real number. • Irreducible quadratic factors of the form + + ax bx c, 2 where a, b, and c are real numbers, ≠ a 0, and − < b ac 4 0. 2 The negative discriminant guarantees that + + ax bx c 2 cannot be written as the product of two linear factors with real coefficients. As it turns out, there are four cases to be examined. For all four cases, the expression P Q is proper and in lowest terms. We begin with the case for which Q has only nonrepeated linear factors. NOTE Instead of using subscripts, such as A1 and A2 in equation (1), it is common to use A and B to represent the unknown numbers. j Decomposing P Q where Q Has Only Nonrepeated Linear Factors Find the partial fraction decomposition of − + x x x5 6 . 2 Solution EXAMPLE 2 First, factor the denominator, ( )( ) − + = − − x x x x 5 6 2 3 2 and notice that the denominator contains only nonrepeated linear factors. Next, decompose the rational expression according to equation (1): − + = − + − x x x A x B x 5 6 2 3 2 (2) We need to find the values of A and B that make this identity true.To find A and B, clear the fractions by multiplying both sides by ( )( ) − − = − + x x x x 2 3 5 6. 2 The result is ( ) ( ) ( ) ( ) = − + − = + + − − x A x B x x A B x A B 3 2 3 2 (3) 1 Decompose P Q where Q Has Only Nonrepeated Linear Factors Case 1: Q has only nonrepeated linear factors. Under the assumption that Q has only nonrepeated linear factors, the polynomial Q has the form ( ) ( )( ) ( ) = − − − Q x x a x a x a · · n 1 2 where no two of the numbers a a a , , . . . , n 1 2 are equal. In this case, the partial fraction decomposition of P Q is of the form ( ) ( ) = − + − + + − P x Q x A x a A x a A x a n n 1 1 2 2 (1) where the numbers A A A , , . . . , n 1 2 are to be determined. Combine like terms.
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