In any field, does $c$ is a root of a polynomial $ f(x) ∈ F[x]$ implies that $f(x)=(x-c)g(x)$?

Question

I know the statement holds in the field R and C

But for any other fields, does it hold in general? If so, how can I formally prove that. Also I think there is something related to zero divisors, if any, in the ring $F[x]$. It seems subtle...

Could some one formally prove or give me some counterexample of the statement? And could someone point out if there exist some subtle relation to the zero divisors in $F[x]$?

Many thanks?

Answer 1

Yes, this is because the ring of polynomials with coefficients in a field is a euclidean domain under the degree function. In other words $f(x)=(x-\alpha)g(x)+b$, with $b\in F$.

Cleary $(x-\alpha$ divides $f(x)$ if and only if $b=0$ if and only if $f(\alpha)=0$