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)! .

Printable View

- August 30th 2012, 12:09 PMbrucewayneProving 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)! .

- August 31st 2012, 10:51 AMVlasevRe: Proving Divisibility
This seems like an interesting question. What have you tried so far?

- August 31st 2012, 07:18 PMBingkRe: Proving Divisibility
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... - September 7th 2012, 11:37 AMbrucewayneRe: Proving Divisibility
I was thinking along the lines of proving by induction, but I'm not really sure.