816 CHAPTER 11 Systems of Equations and Inequalities Now write ( ) + − + = + − + − x x x x A x B x C x 2 2 1 1 3 2 2 (4) Again, clear fractions by multiplying both sides by ( ) − + = − x x x x x 2 1 . 3 2 2 The result is the identity ( ) ( ) + = − + − + x A x Bx x Cx 2 1 1 2 (5) Let = x 0 in this identity and the terms containing B and C drop out, leaving ( ) = − A 2 1 , 2 or = A 2. Similarly, let = x 1, and the terms containing A and B drop out, leaving = C 3 . Then equation (5) becomes ( ) ( ) + = − + − + x x Bx x x 2 2 1 1 3 2 Let = x 2 (any number other than 0 or 1 will work as well). The result is = ⋅ + ⋅ ⋅ + ⋅ = + + = − = − B B B B 4 2 1 2 1 3 2 4 2 2 6 2 4 2 2 Therefore, = = − A B 2, 2, and = C 3. From equation (4), the partial fraction decomposition is ( ) + − + = + − − + − x x x x x x x 2 2 2 2 1 3 1 3 2 2 TIP The strategy for choosing values of x is to zero out factors and obtain an identity that has only one unknown. j NOTE x2 is a repeated linear factor, since ( )( ) = − − x x x 0 0 . 2 j Decomposing P Q where Q Has Repeated Linear Factors Find the partial fraction decomposition of ( ) − − x x x 8 1 . 3 2 3 Solution EXAMPLE 4 The denominator contains the repeated linear factors x2 and ( ) −x 1 . 3 The partial fraction decomposition has the form ( ) ( ) ( ) − − = + + − + − + − x x x A x B x C x D x E x 8 1 1 1 1 3 2 3 2 2 3 (6) As before, clear fractions and obtain the identity ( ) ( ) ( ) ( ) −= −+ −+ −+ −+ x Ax x B x Cx x Dx x Ex 8 1 1 1 1 3 3 3 2 2 2 2 (7) Let = x 0. (Do you see why this choice was made?) Then ( ) − = − = B B 8 1 8 Let = x 1 in equation (7). Then − = E 7 Notice that there are no other values of x we can choose to zero out factors in equation (7). To find the other unknown values, we must now use a different technique. Let = B 8 and = − E 7 in equation (7), and collect like terms. [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) −= −+ −+ −+ −− −− −+−+ = −+ −+ − − + − = − − + − + − − + = − − + − + − + = − + − + x Ax x x Cx x Dx x x x x x x x Ax x Cx x Dx x x x x x x A x Cx x Dx x x x x x A x Cx x Dx x A x Cx x Dx 8 1 8 1 1 1 7 8 8 3 3 1 7 1 1 1 7 31 24 1 1 1 1 7 24 1 1 1 7 24 1 1 3 3 3 2 2 2 2 3 3 2 2 3 2 2 2 3 2 2 2 2 (8)
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