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Thread: Partial fraction decomposition

  1. #1
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    Partial fraction decomposition

    $\displaystyle \frac{1}{x^3(x^2+1)}=\frac{ax^2+bx+c}{x^3}+\frac{d x+e}{x^2}\\$
    $\displaystyle ax^4+bx^3+cx^2+dx^4+ex^3=x^4(a+d)+x^3(b+e)+cx^2=1\ \$
    $\displaystyle c=0\\$
    $\displaystyle b+e=0\\$
    $\displaystyle a+d=0\\$
    i got 4 variables with 2 equations
    what to do??
    Last edited by mr fantastic; Aug 17th 2009 at 01:21 PM.
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  2. #2
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    Hi transgalactic

    Your partial fraction is wrong

    $\displaystyle
    \frac{1}{x^3(x^2+1)}\neq\frac{ax^2+bx+c}{x^3}+\fra c{dx+e}{x^2}\\
    $ because if we put the denominator of RHS together, it won't become $\displaystyle x^3(x^2+1)$ but $\displaystyle x^3$ (different from LHS)

    It should be :
    $\displaystyle
    \frac{1}{x^3(x^2+1)}=\frac{A}{x}+\frac{B}{x^2}+\fr ac{C}{x^3}+\frac{Dx+E}{x^2+1}$
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  3. #3
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    Quote Originally Posted by transgalactic View Post
    $\displaystyle \frac{1}{x^3(x^2+1)}=\frac{ax^2+bx+c}{x^3}+\frac{d x+e}{x^2}\\$
    $\displaystyle ax^4+bx^3+cx^2+dx^4+ex^3=x^4(a+d)+x^3(b+e)+cx^2=1\ \$
    $\displaystyle c=0\\$
    $\displaystyle b+e=0\\$
    $\displaystyle a+d=0\\$
    i got 4 variables with 2 equations
    what to do??
    $\displaystyle \frac{1}{x^3(x^2+1)} = \frac{a}{x} + \frac{b}{x^2} + \frac{c}{x^3} + \frac{dx+e}{x^2+1}$

    $\displaystyle 1 = ax^2(x^2+1) + bx(x^2+1) + c(x^2+1) + (dx+e)x^3
    $

    $\displaystyle 1 = (a+d)x^4 + (b+e)x^3 +(a+c)x^2 + bx + c$

    equating coefficients ...

    $\displaystyle a+d = 0$

    $\displaystyle b+e = 0$

    $\displaystyle a+c = 0$

    $\displaystyle b = 0$

    $\displaystyle c = 1$


    from the above equations ...

    $\displaystyle a = -1$ , $\displaystyle b = 0$ , $\displaystyle c = 1$ , $\displaystyle d = 1$ , $\displaystyle e = 0$

    $\displaystyle \frac{1}{x^3(x^2+1)} = -\frac{1}{x} + \frac{1}{x^3} + \frac{x}{x^2+1}$
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  4. #4
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    can you give me a link to the manual of this stuff
    in order to know thurely all the laws
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  5. #5
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    Quote Originally Posted by transgalactic View Post
    can you gove me a link to this stuff
    in order to know thurely all the laws
    partial fraction decomposition - Google Search
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  6. #6
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    thanks
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