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Math Help - Find all polynomials p(x)

  1. #1
    Junior Member
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    Question Find all polynomials p(x)

    Hi so I was doing this question on my own for a while now, and I really can't find a way to answer part b)
    I looked everywhere in the textbook nothing similar.
    do you guys think the question is missing something, or am i missing my brain!

    here it is
    Find all polynomials p(x)-last-question.jpg

    Please help thank you
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  2. #2
    MHF Contributor

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    Tejas
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    Re: Find all polynomials p(x)

    first, we have to see how this relates to the matrix A.

    what does the matrix A do to a polynomial p(x) = ax2+bx+c = [a,b,c] in the basis {x2,x,1}?

    it takes [a,b,c] to [a-b+c,a+b+c,25a+5b+c].

    note that p(-1) = a(-1)2 + b(-1) + c = a-b+c

    p(1) = a+b+c

    p(5) = a(52) + b(5) + c = 25a+5b+c.

    so A takes p(x) to q(x) = p(-1)x2 + p(1)x + p(5).

    suppose we are just given q(x), then to find p(x) we take A-1(q(x)).

    we are told by the problem that q(x) = -x2+3x+31, which in our basis is [-1,3,31]. find A-1 of this vector.

    (the fact that A is invertible means there is only ONE polynomial p(x) in P2 with this property).
    Thanks from ohYeah
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  3. #3
    Junior Member
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    Re: Find all polynomials p(x)

    what can i say Deveno your always there to help

    thank you very much

    your explanations are always perfect
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