# Thread: Finding the largest factor for a polynomial

1. ## 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 $\displaystyle n^5 - 5n^3 + 4n$ for all whole numbers $\displaystyle n$?

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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:

$\displaystyle n(n-1)(n+1)(n-2)(n+2)$.

Any insight would be appreciated!

Thanks.

2. 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.