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Thread: partial fractions cubic base

  1. #1
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    partial fractions cubic base

    i still cant get the correct answer for this,i simply equated coefficients etc anad got as far as the first two terms in the answer. The question is basically
    Q express $\displaystyle \frac{x^5-1}{x^2(x^3+1)}$ in partial fractions ?[/quote]$\displaystyle \frac{x^5-1}{x^5+x^2}$
    supposed working,,
    $\displaystyle \frac{x^5-1+x^2-x^2}{x^5+x^2}$

    $\displaystyle \frac{x^5+x^2}{x^5+x^2}-\frac{1+x^2}{x^5+x^2}$

    $\displaystyle 1-\frac{1+x^2}{x^2(x^3+1)}$

    I'd start it that way. You can try and finish it.

    In this case:

    $\displaystyle 1+x^2\equiv(Ax+B)(x^3+1)+(Cx^2+Dx+E)(x^2)$

    Since this is an identity, it is true for all values of x (this is a great help in determing the values of A,B,C,D and E!).[/quote] i miust be doing something awfully wrong
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  2. #2
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    You can make the last part a bit simpler by factorising $\displaystyle 1+x^3$ as $\displaystyle (1+x)(1-x+x^2)$. Then you'll want $\displaystyle \frac{1+x^2}{x^2(1+x)(1-x+x^2)} = \frac{Ax+B}{x^2} + \frac C{1+x} + \frac{Dx+E}{1-x+x^2}$.
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  3. #3
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    Hi, thanks for your reply and your method works but i can't help to not understand why it doesnt work when i use the partial fractions,$\displaystyle \frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}+\frac{Dx+E }{x^2-x+1}$
    Using this u can quickly deduce B=1,,,and C=2/3 by elimiting their factors but then when i put in x=1,,i get -2/3=2(A+D+E)
    $\displaystyle (x^4)$,,,E=-1/3
    $\displaystyle (x^5)$,,,A+D=-2/3 which then gives me(putting these together):2A+2D=0

    I know A must be zero and D cant be zero as the answer to this question is
    $\displaystyle 1-\frac{1}{x^2}-\frac{2}{3(x+1)}-\frac{1-2x}{3(x^2-x+1)}$
    NB previous post was 1 - p(x)/q(x) IF SOMEONE COULD HELP THEYD BE REALLY COOL!!!!XD
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  4. #4
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    You seem to be doing it mostly correctly. There must be a simple arithmetic error somewhere. Starting from

    $\displaystyle 1+x^2\equiv (Ax+B)(1+x^3)+ Cx^2(1-x+x^2) +(Dx+E)x^2(1+x)$,

    you get B=1 and C=2/3 (by putting x=0 and x=-1). So the identity becomes

    $\displaystyle 1+x^2\equiv (Ax+1)(1+x^3)+ \tfrac23x^2(1-x+x^2) +(Dx+E)x^2(1+x)$.

    Now compare the coefficients of x on both sides, to get A=0, so the identity becomes

    $\displaystyle 1+x^2\equiv 1+x^3 + \tfrac23x^2(1-x+x^2) +(Dx+E)x^2(1+x)$.

    Next, compare the coefficients of $\displaystyle x^4$ on both sides, to get $\displaystyle 0=\tfrac23+D$, so the identity becomes

    $\displaystyle 1+x^2\equiv 1+x^3 + \tfrac23x^2(1-x+x^2) +(\tfrac23x+E)x^2(1+x)$.

    Now you can wrap it up by putting x=1 (say) to get an equation for E.
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  5. #5
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    my bad i got it thanks
    Last edited by oxrigby; Oct 11th 2008 at 02:16 PM.
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