Question

How do you write the partial fraction decomposition of the rational expression `(x^3+2)/(x^2-x)`?

Answer 1

`4/(x-1) - 3/x`

Explanation:

first step is to factor the denominator

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

Since these factors are linear,the numerators will be constants, say A and B.

`rArr (x^3+2)/(x(x-1)) = A/x + B/(x-1)`

multiply through by x(x-1)

`x^3 + 3 = A(x-1) + Bx "..............................................(1)"`

The aim now is to find the values of A and B. Note tat if x = 1 the term with A will be zero and if x = 0 the term with B will be zero.
This is the starting point for finding A and B.

let x = 1 in (1) : 4 = B

let x = 0 in (1) :`3 = - A rArr A = - 3`

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