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)! .
Just some ideas ...
The integer root is very important. You can represent a polynomial as where the are integers (as stated in the problem). How would you represent the polynomial if you knew one of its roots, say ?
What would happen to the product if (i.e. is also nonnegative) and ? what about if and ? Remember, is an integer root of the polynomial, and is nonnegative.
What about similar situations for when (i.e. what are the possible cases if is negative?
That should help you get started...