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Partial fraction decomposition using matrix

  1. Jun 25, 2016 #1
    1. The problem statement, all variables and given/known data
    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.
    2. Relevant equations
    (11x^2 - 5x - 10) / (5x^3 - 5x^2)

    3. 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: Jun 25, 2016
  2. jcsd
  3. Jun 25, 2016 #2

    ehild

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    It is correct so far. What did you get for A,B,C?
     
  4. Jun 25, 2016 #3
    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!
     
  5. Jun 26, 2016 #4

    haruspex

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    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.
     
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