How do you express (x) / (x^(3)-x^(2)-2x +2) in partial fractions?

1 Answer
Mar 5, 2016

x/(x^3−x^2−2x+2)hArrx/(x^2-2)-1/(x-1)

Explanation:

To express x/(x^3−x^2−2x+2) in partial fractions, we should first factorize (x^3−x^2−2x+2).

(x^3−x^2−2x+2) = x^2(x-1)-2(x-1)=(x^2-2)(x-1)

Hence, x/((x^2-2)(x-1))hArr((Ax+B)/(x^2-2))+C/(x-1)

The latter can be expanded as ((Ax+B)(x-1)+C(x^2-2))/((x^2-2)(x-1)) or

(Ax^2+Bx-Ax-B+Cx^2-2C)/((x^2-2)(x-1)) or

((A+C)x^2+(B-A)x-B-2C)/((x^2-2)(x-1)) and as it is equal to x/(x^3−x^2−2x+2), we have

A+C=0, B-A=1 and -B-2C=0

Eliminating C from first and third equation, by multiplying first by 2 and adding it to third, we get 2A-B=0.

Adding this to second, we get A=1. This gives us B=0 and then C=-1.

Hence x/(x^3−x^2−2x+2)hArrx/(x^2-2)-1/(x-1)