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

But this can't be correct because leads to wrong answers.
Thank you!

 

[ehild]

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?

 

[ducmod]

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]

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.