# monic polynomials

• April 28th 2009, 05:55 PM
hka210
monic polynomials
Any ideas on this one?

Let f(x) be a monic polynomial over Z. Show that if n is a nonzero integer such that f(n) = 0, then n divides f(0).

Thanks!
• April 28th 2009, 07:42 PM
Gamma
$f(x)=x^k + a_{k-1}x^{k-1} + ... + a_1x + a_0$
$f(n)=n^k + a_{k-1}n^{k-1} + ... + a_1n + a_0=0$
$n^k + a_{k-1}n^{k-1} + ... + a_1n =- a_0$
n divides the left side, so n divides $a_0=f(0)$
• April 29th 2009, 04:26 PM
HallsofIvy
Quote:

Originally Posted by hka210
Any ideas on this one?

Let f(x) be a monic polynomial over Z. Show that if n is a nonzero integer such that f(n) = 0, then n divides f(0).

Thanks!

If f(n)= 0, then (x- n) is a factor of f(x).