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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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