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Math Help - Polynomial Degree Problem

  1. #1
    Junior Member PaulAdrienMauriceDirac's Avatar
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    Polynomial Degree Problem

    Hello, this problem is again from Gelfand and Shen's Algebra textbook.

    Problem 171: The highest coefficient of P(x) is 1, and we know that P(1)=0, P(2)=0, P(3)=0,\ldots,P(9)=0,P(10)=0. What is the minimal possible degree of P(x)? Find P(11) for this case.

    Any hints would be appreaciated.
    Last edited by PaulAdrienMauriceDirac; November 13th 2012 at 10:42 AM.
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  2. #2
    Junior Member PaulAdrienMauriceDirac's Avatar
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    Re: Polynomial Degree Problem

    I solved it, I don't know why I didn't see it, this was very easy.

    Since P(1)=0, P(2)=0, P(3)=0,\ldots,P(9)=0,P(10)=0., this means that P(x) has at least 10 roots so it is at least degree 10 polynomial.

    P(x)=Q(x-1)(x-2) \ldots (x-9)(x-10), Q has to be 1 since maximum coefficient is 1, so

    P(x)=(11-1)(11-2) \ldots (11-9)(11-10)=10!=3628800.
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