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Math Help - Getting certain coefficient from a polynomial

  1. #1
    Newbie Mobius's Avatar
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    Getting certain coefficient from a polynomial

    Say I have a regular polynomial like 2x^3+5x^2+7x or (x+x^2)^3.

    Is there a way to extract the coefficient for the nth degree?

    For example, if f(x)=4x^3+5x^2-6x and I need the coefficient for x^2 (which is 5 in this case), is there some magic function g(f,n) which does:

    g(f(x),2) = 5

    ?

    (and similarly, g(f(x),3) would be 4, etc)
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  2. #2
    MHF Contributor
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    Only if your polynomial is written as a binomial, i.e. \displaystyle (ax+b)^n and \displaystyle n is a positive integer.

    The binomial expansion is \displaystyle (ax+b)^n = \sum_{r = 0}^n{{n\choose{r}}(ax)^{n-r}b^r}, where \displaystyle {n\choose{r}} = \frac{n!}{r!(n-r)!}.

    So each coefficient of \displaystyle x^{n-r} will be \displaystyle {n\choose{r}}a^{n-r}b^r.
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  3. #3
    Newbie Mobius's Avatar
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    I see, thanks. So there's no trick to extract coefficients from any polynomial in general? (I was thinking maybe something smart with modulo x^{n+1} or something, but didn't really get anywhere)

    How about the specific case where the polynomial looks like (x+x^2+x^3+\cdots+\x^n)^r ? Any clever possibilities there?
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