# Partial fraction decomposition using matrix

## 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!

Last edited:

ehild
Homework Helper

## 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!
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
And the answer is:
3/x + 2/x^2 - 4 / 5(x - 1)

Thank you!

haruspex
Science Advisor
Homework Helper
Gold Member
2020 Award
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
And the answer is:
3/x + 2/x^2 - 4 / 5(x - 1)

Thank you!
From the equations you had, you should have got C=-4/5. That would have given the right answer since e.g. your 1/x term is defined as A/(5x)=15/(5x) = 3/x. Similarly the B term.
You could have made life a little simpler by taking out the 1/5 as a common factor up front, only bringing it back right at the end. You certainly did not need to include it in the decomposition; you could have decomposed as A/x+ etc.