# Partial fraction decomposition using matrix

• ducmod

## 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

Thank you!

Last edited:

## 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

Thank you!
It is correct so far. What did you get for A,B,C?

It is correct so far. What did you get for A,B,C?
thank you very much for taking time to check this. I was panicking a bit, because it seemed so wrong and I couldn't figure out why.
A = 15, B = 10, C = -4
3/x + 2/x^2 - 4 / 5(x - 1)

Thank you!

thank you very much for taking time to check this. I was panicking a bit, because it seemed so wrong and I couldn't figure out why.
A = 15, B = 10, C = -4