Hi, I am not sure of a certain question and would like someone to explain a few part of the solution to me, here is the question and the solution:

Express 5x^2 - 8x - 5 / (x - 1)(x^2 - 1) in partial fractions

Here is the solution:

Factorizing the denominator first:

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

Let 5x^2 - 8x - 5 / (x - 1)(x^2 - 1) = **A / x + 1 + B / X - 1 + C / (X - 1)^2**

Multiply both sides by (X + 1)(X - 1)^2 to remove the denominators of both sides

5x^2 - 8x - 5 = A(x - 1)^2 + B(x + 1)(x - 1) + C(x + 1) ...............(1)

Put X = 1 into (1)

5(1)^2 - 8(1) - 5 = A(1 - 1)^2 + B(1 + 1)(1 - 1) + C(1 + 1)

C = -4 --------------------(2)

and so on....

my question is the one in bold and the factorization,

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

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

I am not sure why is there another x - 1 since I have workout:

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

therefore (x - 1)(x^2 -1) which is the original denominator in that question and when I factorize out, the answer is (x - 1)(x - 1)(x + 1) which is the same as (x + 1)(x - 1)^2

for the partial fraction rules, I have to write it as

5x^2 - 8x - 5 / (x - 1)(x^2 - 1) = A / x + 1 + **B / x - 1** + C / (x - 1)^2

so I am not sure where does this B / x -1 in the denominator comes from? can anyone explain to me?

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