818 CHAPTER 11 Systems of Equations and Inequalities Now Work Example 5 using equation (10) and assigning values to x. Now Work PROBLEM 25 Decomposing P Q where Q Has a Repeated Irreducible Quadratic Factor Find the partial fraction decomposition of ( ) + + x x x 4 . 3 2 2 2 EXAMPLE 6 Notice the nonrepeated linear factor −x 1 and the nonrepeated irreducible quadratic factor + + x x 1. 2 Allow for the term − A x 1 by Case 1, and allow for the term + + + Bx C x x 1 2 by Case 3. Then − − = − + + + + x x A x Bx C x x 3 5 1 1 1 3 2 (9) Multiply both sides of equation (9) by ( ) ( ) − = − + + x x x x 1 1 1 3 2 to obtain the identity ( ) ( )( ) − = + + + + − x A x x Bx C x 3 5 1 1 2 (10) Expand the identity in (10) and combine like terms to obtain ( ) ( ) ( ) − = + + − + + − x A B x A B C x A C 3 5 2 Equating coefficients of like powers leads to the system of equations + = − + = − =− ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ (1) (2) (3) A B A B C A C 0 3 5 The solution of this system is = − = = A B C 2 3 , 2 3 , 13 3 . Then, from equation (9), − − = − − + + + + x x x x x x 3 5 1 2 3 1 2 3 13 3 1 3 2 Case 4: Q contains a repeated irreducible quadratic factor. Suppose the polynomial Q contains a repeated irreducible quadratic factor ( ) + + ≥ ax bx c n, 2, n 2 n an integer, and − < b ac 4 0. 2 Then, in the partial fraction decomposition of P Q , allow for the terms ( ) ( ) + + + + + + + + + + + + A x B ax bx c A x B ax bx c A x B ax bx c n n n 1 1 2 2 2 2 2 2 where the numbers A B A B A B , , , , . . . , ,n n 1 1 2 2 are to be determined. 4 Decompose P Q where Q Has a Repeated Irreducible Quadratic Factor
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