# How do you write the partial fraction decomposition of the rational expression #x^2 / ((x-1)^2 (x+1))#?

##### 1 Answer

Feb 9, 2016

# 3/4(x-1) + 1/2(x-1)^2 + 1/4(x+1) #

#### Explanation:

firstly note that the factors of

#(x-1)^2 # are (x-1) and

# (x-1)^2# hence

# x^2/((x-1)^2(x+1)) = A/(x-1) + B/(x-1)^2 + C/(x+1) # now multiply through by

# (x-1)^2(x+1) #

# x^2 = A(x-1)(x+1 ) + B(x+1) + C(x-1)^2 ......(1)# now require to find values for A , B and C . Note that if x = 1 , the terms with A and C will be zero and if x = -1 the terms with A and B will be zero. This is the starting point for finding values.

let x = 1 in (1) : 1 = 2B

# rArr B = 1/2 # let x = -1 in(1) : 1 = 4C

# rArr C = 1/4 # can now choose any value of x , to substitute into equation (1)

let x = 0 in (1) : 0 = -A + B + C

hence A = B + C # = 1/2 + 1/4 = 3/4

# rArr x^2/((x-1)^2(x+1)) = 3/4(x-1) + 1/2(x-1)^2 + 1/4(x+1) #