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Math Help - Proving Divisibility

  1. #1
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    Proving Divisibility

    The polynomial f(X) has integer coefficients and an integer root. Prove that, for every non negative integer n, the product f(0)f(1)...f(n) is divisible by (n+1)! .
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    Re: Proving Divisibility

    This seems like an interesting question. What have you tried so far?
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    Re: Proving Divisibility

    Just some ideas ...

    The integer root is very important. You can represent a polynomial as f(x) = \sum_{i=0}^n a_i x^i where the a_i are integers (as stated in the problem). How would you represent the polynomial if you knew one of its roots, say r?


    What would happen to the product f(0)f(1) ... f(n) if r \geq 0 (i.e. r is also nonnegative) and n \geq r? what about if r \geq 0 and  n < r? Remember, r is an integer root of the polynomial, and n is nonnegative.
    What about similar situations for when r < 0 (i.e. what are the possible cases if r is negative?

    That should help you get started...
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    Re: Proving Divisibility

    I was thinking along the lines of proving by induction, but I'm not really sure.
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