I came across this partial fractions problem yesterday and I'm confused as to why I get the wrong answer although the factorising seems correct to me. I'll list down what I did and what is the answer provided by the book.

__The steps I took to break it up__

x / (1-x^2)(x-1)

= x / (1+x)(1-x)(x-1)

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

= - [1/4(x-1)] - [1/2(x-1)^2] + [1/4(1+x)]

However, the book states that the answer is [1/4(1-x)] - [1/2(1-x)^2] + [1/4(1+x)].

When I tested my answer and the book's, I found out that the book's answer gave the fraction that I was asked to break up.

Is there a reason why I must factorise the denominator into (1-x)^2 (1+x) and not (x-1)^2 (1+x)?

Thanks in advance.