SECTION 11.5 Partial Fraction Decomposition 817 Decomposing P Q where Q Has a Nonrepeated Irreducible Quadratic Factor Find the partial fraction decomposition of − − x x 3 5 1 . 3 Solution EXAMPLE 5 Now work with equation (8). Let = x 0. Then = A 24 Let = x 1 in equation (8). Then = D 17 Use = A 24 and = D 17 in equation (8). ( ) ( ) − + = − + − + x x Cx x x 7 24 24 1 1 17 2 Let = x 2 and simplify. ( ) − + = + + − = − = C C C 14 24 24 2 34 48 2 24 The numbers A, B, C, D, and E are now known. So, from equation (6), ( ) ( ) ( ) − − = + + − − + − + − − x x x x x x x x 8 1 24 8 24 1 17 1 7 1 3 2 3 2 2 3 Now Work Example 4 by solving the system of five equations containing five variables that results by expanding equation (7) . Now Work PROBLEM 23 The final two cases involve irreducible quadratic factors. A quadratic factor is irreducible if it cannot be factored into linear factors with real coefficients. A quadratic expression + + ax bx c 2 is irreducible whenever − < b ac 4 0. 2 For example, + + x x 1 2 and + x 4 2 are irreducible. Case 3: Q contains a nonrepeated irreducible quadratic factor. Suppose Q contains a nonrepeated irreducible quadratic factor of the form + + ax bx c. 2 Then, in the partial fraction decomposition of P Q , allow for the term + + + Ax B ax bx c 2 where the numbers A and B are to be determined. Factor the denominator, ( ) ( ) − = − + + x x x x 1 1 1 3 2 (continued) 3 Decompose P Q where Q Has a Nonrepeated Irreducible Quadratic Factor
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