Stitz-Zeager_College_Algebra_e-book

1 sequences 553 3 from 2n 1 we have that an 2n 1

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Unformatted text preview: Dx + E =22 =+ +2 4 + x2 2 x x (x + 1) x x x +1 1 2 Recall this means it has no real zeros; see Section 3.4. Recall this means x = 0 is a zero of multiplicity 2. 8.6 Partial Fraction Decomposition 523 C C However, if we look more closely at the term Bx+C , we see that Bx+C = Bx + x2 = B + x2 . The x x2 x2 x2 B A term x has the same form as the term x which means it contributes nothing new to our expansion. Hence, we drop it and, after re-labeling, we ﬁnd ourselves with our new guess: x2 − x − 6 x2 − x − 6 A B Cx + D =22 = + 2+ 2 4 + x2 x x (x + 1) x x x +1 Our next task is to determine the values of our unknowns. Clearing denominators gives x2 − x − 6 = Ax x2 + 1 + B x2 + 1 + (Cx + D)x2 Gathering the like powers of x we have x2 − x − 6 = (A + C )x3 + (B + D)x2 + Ax + B In order for this to hold for all values of x in the domain of f , we equate the coeﬃcients of corresponding powers of x on each side of the equation3 and obtain the system of linear equations (E 1) A + C (E 2) B...
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