I am doing a chapter on partial fraction decomposition, and it seems I do not understand it correctly.
Here is the exercise doing which I get wrong answers. Please, take a look at the way I proceed and, please, let me know what is wrong in my understanding.
(11x^2 - 5x - 10) / (5x^3 - 5x^2)
The Attempt at a Solution
First, I look at the denominator and see that there are three factors:
5x^2 ( x - 1) =>
5x and 5x^2 and (x - 1)
Second, I construct the form to begin the partial fraction decomposition:
(11x^2 - 5x - 10) / (5x^3 - 5x^2) = A / 5x + B / 5x^2 + C / (x - 1)
Here is the question: am I on the right path, and do I use the constant term 5 correctly in the denominator?
Third step: I eliminate the denominator on both sides:
(11x^2 - 5x - 10) = A 5x^2 (x - 1) / 5x + B 5x^2 (x - 1) / 5x^2 + C 5x^2 (x - 1) / (x - 1)
(11x^2 - 5x - 10) = A x (x - 1) + B (x - 1) + C 5x^2
11x^2 - 5x - 10 = Ax^2 - Ax + Bx - B + C 5x^2
11x^2 - 5x - 10 = x^2(A + 5C) + x (B - A) - B
Fourth step: I create a matrix:
A + 5C = 11
B - A = - 5
- B = - 10
But this can't be correct because leads to wrong answers.