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decompose 5x+1/x^2-x-12 into partial functions?

  • calculus -

    I think you mean partial fractions. That is what this method was called when I went to school. First factor the denominator.

    (5x+1)/(x^2-x-12) = (5x+1)/[(x-4)(x+3)]

    This can be written as
    A/(x-4) + B/(x+3) where A and B are certain constants. Here is how you can solve for A and B.

    Require that
    (5x+1)/[(x-4)(x+3)] = A/(x-4) + B/(x+3)
    = [A(x+3) +B(x-4)]/[(x-4)(x+3)]
    = (A+B)x +3A-4B
    This will be true if A+B = 5 and 3A-4B = 1
    4A+4B = 20
    Solve for A and B
    3A+3B = 15
    3A-4B = 1
    7B = 14
    B = 2 A = 3
    So the answer is
    3/(x-4) + 2/(x+3)

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