Expert: Sherman D. - 6/30/2006

Question
Find A,B,and C such that :

(x-1)/(x+1)(x-2)^2=A/(x+1)+ B/(x-2)+C/(x-2)^2

Answer
((x - 1)/((x + 1)(x - 2)^2))) = (A/(x + 1)) + (B/(x - 2)) + (C/(x - 2))^2

becomes

x - 1 = A(x - 2)^2 + B(x + 1)(x - 2) + C(x + 1)

-1 - 1 = A(-1 - 2)^2
-1 + (-1) = A(-1 + (-2))^2
-2 = A(-3)^2
-2 = 9A
A = (-2/9)

2 - 1 = C(2 + 1)
1 = 3C
C = (1/3)

0 = A + B
0 = (-2/9) + B
B = (2/9)

so

((x - 1)/((x + 1)(x - 2)^2)) = (-2/(9(x + 1))) + (2/(9(x - 2))) + (1/(3(x - 2)^2))

(-2/(9x + 9)) + (2/(9x - 18)) + (1/(3((x - 2)(x - 2)))
(-2/(9x + 9)) + (2/(9x - 18)) + (1/(3(x^2 - 2x - 2x + 4)))
(-2/(9x + 9)) + (2/(9x - 18)) + (1/(3(x^2 - 4x + 4)))
(-2/(9x + 9)) + (2/(9x - 18)) + (1/(3x^2 - 12x + 12))

Now depending on how you want to type it

ANS :
(-2/(9(x + 1))) + (2/(9(x - 2))) + (1/(3(x - 2)^2))
or
(-2/(9x + 9)) + (2/(9x - 18)) + (1/(3x^2 - 12x + 12))

This website really helped me to figure out how to solve it. www.ncsu.edu/felder-public/kenny/papers/partial.html

Look near the bottom of the site for an example of your problem.