• Support PF! Buy your school textbooks, materials and every day products Here!

Partial fraction decomposition using matrix

  • #1
86
0

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:

Answers and Replies

  • #2
ehild
Homework Helper
15,427
1,827

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?
 
  • #3
86
0
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!
 
  • #4
haruspex
Science Advisor
Homework Helper
Insights Author
Gold Member
32,786
5,050
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.
 

Related Threads on Partial fraction decomposition using matrix

  • Last Post
Replies
7
Views
935
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
3
Views
485
  • Last Post
Replies
17
Views
2K
  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
7
Views
2K
Top