SOLUTION: Hi, i'm really stuck on this problem for partial fractions and would really appreciate the help! Find the partial fraction decomposition of the following rational expression:

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: Hi, i'm really stuck on this problem for partial fractions and would really appreciate the help! Find the partial fraction decomposition of the following rational expression:       Log On


   

Question 447148: Hi, i'm really stuck on this problem for partial fractions and would really appreciate the help!
Find the partial fraction decomposition of the following rational expression:
5x^2+x+5/x^3+x^2+2x+2
Thanks!
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
Find the partial fraction decomposition of the following rational expression:
%285x%5E2%2Bx%2B5%29%2F%28x%5E3%2Bx%5E2%2B2x%2B2%29
Factor the denominator, I used synthetic division
________________
-1|1 + 1 + 2 + 2
and got
%285x%5E2%2Bx%2B5%29%2F%28x%5E2%2B2%29%28x%2B1%29%29 = A%2F%28%28x%5E2%2B2%29%29 + B%2F%28%28x%2B1%29%29 = %28A%28x%2B1%29%2BB%28x%5E2%2B2%29%29%2F%28%28x%5E2%2B2%29%28x%2B1%29%29
If the denominators are equal the numerators equal, so we have:
5x^2 + x + 5 = A(x+1) + B(x^2+2)
:
let x=-1, then one factor will drop out, we can solve for B
5(-1)^2 - 1 + 5 = A(-1+1) + B(-1^2+2)
5 - 1 + 5 = 0 + 3B
9 = 3B
B = 3
:
Replace B with 3, Find A
5x^2 + x + 5 = A(x+1) + 3(x^2+2)
5x^2 + x + 5 = A(x+1) + 3x^2 + 6
5x^2 - 3x^2 + x + 5 - 6 = A(x+1)
2x^2 + x - 1 = A(x+1)
Factor
(2x-1)(x+1) = A(x+1)
Divide both sides by (x+1)
(2x - 1) = A
:
so we have
%285x%5E2%2Bx%2B5%29%2F%28x%5E3%2Bx%5E2%2B2x%2B2%29 = %282x-1%29%2F%28%28x%5E2%2B2%29%29 + 3%2F%28%28x%2B1%29%29