Expert: Steve Holleran - 10/2/2008

Question
Express the following in partial fraction:


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


i have tried it many times for hours, again & again still cant get it right. Much appreciated.

Answer
Hello,

I think it goes like this:


 (x^2-x)/(x^2+3)(x^2+2) =(Ax + B)/(x^2 + 3)+(Cx + D)/(x^2+2)

          x^2 - x = (Ax + B)(x^2+2) + (Cx + D)(x^2 + 3)

     x^2 - x = Ax^3 + Bx^2 + 2Ax + 2B + Cx^3 + Dx^2 + 3Cx + 3D

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

Then  A + C = 0      B+ D = 1    2A - 3C = -1     2B + 3D = 0

Solving the A,C system:

     2A - 3C = -1   ------->   2A - 3C =-1

      A +  C = 0    ----->    -2A -2C = 0
                             -------------

                                 -5C = -1  so C = 1/5  ,then A = -1/5

If you solve the B, D system, you get B = 3 and D = -2

So, the solution is

   (-1/5 x + 3)/(x^2 + 3)   + (1/5 x - 2)/(x^2 + 2)


Hope this helps
Steve