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## Homework Statement

Hello!

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.

## Homework Equations

(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.

Thank you!

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