Question

Write out the partial fraction decomposition of the following rational expression.


1.) 3x^2-7x-2/x^3-x

2.) x^3-x+3/x^2+x-2

Answer 1

I'll do one, and you can do the other. You can check your work at wolframalpha.com

x^3-x = x(x-1)(x+1) so we want

A/x + B/(x-1) + C/(x+1) = (3x^2-7x-2)/(x^3-x)

So, using the common denominator, we have
A(x^2-1) + Bx(x+1) + Cx(x-1) = 3x^2-7x-2

(A+B+C)x^2 + (B-C)x - A = 3x^2-7x-2

If they are identical, then

A+B+C = 3
B-C = -7
-A = -2
So,
A = 2
B = -3
C = 4

and we have

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

check at

http://www.wolframalpha.com/input/?i=partial+fractions+%283x^2-7x-2%29%2F%28x^3-x%29