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Math

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-1/(x^3)-27

or

-1=A(x^2+3x+9)+Bx+C(x-3)

I'm trying to find the partial fraction decomposition. I think A=-1/21, but how do I find the right values to plug in for x to find B and C?

  • Math -

    Do a cross-multiply to put all over a common denominator:

    (Ax+B)/(x^2+3x+9) + C/(x-3)
    = [(Ax+B)(x-3) + C(x^2+3x+9)]/(x^3-27)
    = (Ax^2+(B-3A)x-3B+Cx^2+3Cx+9C)/(x^3-27)
    = [(A+C)x^2 + (B-3A+3C)x + (9C-3B)] / (x^3-27)

    So, for the numerators to be identical, coefficients of like powers must be equal:

    A+C = 0
    B-3A+3C = 0
    9C-3B = -1

    (A,B,C) = (1/27, 6/27, -1/27)

    so, -1/(x^3-27) = (x+6)/(27(x^2+3x+9)) - 1/(27(x-3))

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