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Math Help - Finding the largest factor for a polynomial

  1. #1
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    Finding the largest factor for a polynomial

    Here's a problem I have as written:

    What is the largest whole number that MUST be a factor of n^5 - 5n^3 + 4n for all whole numbers n?

    --

    It isn't clear to me exactly, but I think the question wants an actual number, i.e., NOT something involving n. After finding the value of the expression for different values of n:

    n = 1 ==> 0

    n = 2 ==> 0

    n = 3 ==> 120

    n = 4 ==> 720

    n = 5 ==> 2520

    n = 6 ==> 6720

    I'm inclined to think the answer is 120, but I have no way of proving it or understanding WHY it is 120. Seems like a random number to me.

    Also, note that (if this is helpful), the expression factors to:

    n(n-1)(n+1)(n-2)(n+2).

    Any insight would be appreciated!

    Thanks.
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  2. #2
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    Answering my own question, much later: in its factored form, the expression is really a product of 5 consecutive integers. That means that it is guaranteed to have a factor of 5, a factor of 4, a factor of 3, and a factor of 2.

    5 x 4 x 3 x 2 = 120.

    Not sure anyone else was wondering what the answer to this was, but just in case.
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