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Math Help - Write Expression In Partial Fraction

  1. #1
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    Write Expression In Partial Fraction

    x^2/(x-1)(x+1) = x^2/x^2-1 = 1,-1

    Therefore,
    x^2/(x-1)(x+1) = 1 + -1/(x-1)(x+1)

    -1/(x-1)(x+1) = A/(x-1) + B/(x+1)
    -1 = Ax + A + Bx - B
    -1 = (A+B)x +A - B
    A+B = 0, B = -A
    -1 = A - B = A - (-A) = 2A
    -1 = 2A, A = -1/2,
    B = -(-1/2) = 1/2

    Therefore,
    x^2/(x-1)(x+1) = 1 - 1/2(x-1) + 1/2(x+1)

    But somehow the answer given to me is
    x^2/(x-1)(x+1) = 1 + 1/2(x-1) - 1/2(x+1)

    Not sure which step I doing wrong, please help me on this. Thank you.
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  2. #2
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    Re: Write Expression In Partial Fraction

    Hello, gilagila!

    \frac{x^2}{(x-1)(x+1)}

    We have: . \frac{x^2}{x^2-1} \;=\;\frac{x^2 - 1 + 1}{x^2-1} \;=\;\frac{x^2-1}{x^2-1} + \frac{1}{x^2-1} \;=\;1 + \frac{1}{x^2-1}

    Then: . \frac{1}{(x-1)(x+1)} \;=\;\frac{A}{x-1} + \frac{B}{x+1}

    . . . . . . 1 \;=\;A(x+1) + B(x-1)


    Let x = 1\!:\;\;1 \;=\;A(2) + B(0) \quad\Rightarrow\quad A \,=\,\tfrac{1}{2}

    Let x = \text{-}1\!:\;\;1 \;=\;A(0) + B(\text{-}2) \quad\Rightarrow\quad B \,=\,\text{-}\tfrac{1}{2}


    Therefore: . \frac{x^2}{(x-1)(x+1)} \;=\; 1 + \frac{\frac{1}{2}}{x-1} - \frac{\frac{1}{2}}{x+1}

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  3. #3
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    Re: Write Expression In Partial Fraction

    Quote Originally Posted by Soroban View Post
    \frac{x^2}{x^2-1} \;=\;\frac{x^2 - 1 + 1}{x^2-1} \;
    Just want to check, how you change to this ??
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  4. #4
    Member sbhatnagar's Avatar
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    Re: Write Expression In Partial Fraction

    Quote Originally Posted by gilagila View Post
    Just want to check, how you change to this ??
    \frac{x^2}{x^2-1}=\frac{x^2-1+1}{x^2-1}

    He subtracted 1 from the numerator and at the same time added it. This does not make any "change" to expression.

    =
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  5. #5
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    Re: Write Expression In Partial Fraction

    oic, thanks for explanation
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