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Math Help - Notate Coefficients of a Polynomial

  1. #1
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    Notate Coefficients of a Polynomial

    Hi,

    Not sure If this thread is in the right part of the forum.

    Anyway, quite simply if one has a polynomial as below,

    Can the coefficents of an arbitrary polynomial be notated as a series or set?

    (x+1)(x^4 + 2x^3 + 2x^2 + 2x + 1) \:=\:x^5 + 3x^4 + 4x^3 + 4x^2 + 3x + 1

    i.e. from this example i would need...

    s = [1, 3, 4, 4, 3, 1]

    If so how?

    Thanks Alex
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  2. #2
    Senior Member Danneedshelp's Avatar
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    I dont think this would fall under the catagory of discrete mathematics.

    I am not sure exactly what you want, but the first thing that came to my mind was this...

    let S=\{1,x,x^{2},...,x^{n-1},x^{n}\}. This set spans \mathbb{R}^{n} because any polynomial fucntion p(x)=a_{0}+a_{1}x+...+a_{n-1}x^{n-1}+a_{n}x^{n} in \mathbb{R}^{n} can be written as,

    p(x)=a_{0}(1)+a_{1}(x)+...+a_{n-1}(x^{n-1})+a_{n}(x^{n})

    So, in your example, your basis is the set S=\{1,x,x^{2},x^{3},x^{4},x^{5}\} in \mathbb{R}^{5}. So any polynimial of degree 5 can be created by these vectors. Since you have coeficients \{1,3,4,4,3,1\} we can write p(x)=1(1)+3(x)+4(x^{2})+4(x^{3})+3(x^{4})+1(x^{5}) which is your polynomial. Thus any polynomial or the form p(x)=a_{0}+a_{1}x+...+a_{5}x^{5} can be created from the set S. The only thing that changes are your coeficients (a_{0},a_{1},...,a_{5}), so I guess you could just list them in a set.

    Thats my stab at it.
    Last edited by Danneedshelp; July 31st 2009 at 11:25 AM.
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