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Math Help - Quick question involving long division

  1. #1
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    Quick question involving long division

    \int \frac {x^3 + 13} {x^2 + 5x + 6} dx

    The first step is to do long division. Can anyone tell me if (x+5) + \frac {-19x - 17} {x^2 + 5x + 6} is the right answer?
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    Quote Originally Posted by Archduke01 View Post
    \int \frac {x^3 + 13} {x^2 + 5x + 6} dx

    The first step is to do long division. Can anyone tell me if (x+5) + \frac {-19x - 17} {x^2 + 5x + 6} is the right answer?
    If you make a common denominator, do you will back to the original fraction?
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    Quote Originally Posted by Miss View Post
    If you make a common denominator, do you will back to the original fraction?
    Sorry? I don't understand your question.
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  4. #4
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    Quote Originally Posted by Archduke01 View Post
    \int \frac {x^3 + 13} {x^2 + 5x + 6} dx

    The first step is to do long division. Can anyone tell me if (x+5) + \frac {-19x - 17} {x^2 + 5x + 6} is the right answer?
    Hi Archduke,

    you can check it quickly...

    x^3+13=\left(x^2+5x+6\right)(x+a)+Rem

    =x^3+5x^2+6x+ax^2+5ax+6a+Rem

    =x^3+(5+a)x^2+(6+5a)x+6a+Rem

    Therefore a=-5 as x^3+13 has no x^2 term.

    It also has no x term, so we cancel it in the remainder

    x^3+13=x^3-19x-30+19x+43

    \frac{x^3+13}{x^2+5x+6}=(x-5)+\frac{19x+43}{x^2+5x+6}
    Last edited by Archie Meade; March 11th 2010 at 11:58 PM. Reason: typo
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