# Proving Divisibility

• Aug 30th 2012, 12:09 PM
brucewayne
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)! .
• Aug 31st 2012, 10:51 AM
Vlasev
Re: Proving Divisibility
This seems like an interesting question. What have you tried so far?
• Aug 31st 2012, 07:18 PM
Bingk
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?