Arithmetic of Polynomials

Question

if $k$ is a field and $f,g,h$ are polynomials such that $f=g+h$, then a polynomial that divides two of the three will divide the third.

I'm having trouble showing doing the problem as polynomials. I know how to do it is $f,g,h$ are integers.

Answer 1

Do it the usual way (as for integers):

If $p$ is a polynomial over $k$ dividing both $g,h$ this means that $\frac{g}{p}$, $\frac{h}{p}$ are also polynomials over $k$. Thus: $$ \frac{f}{p}=\frac{g+h}{p}=\frac{g}{p}+\frac{h}{p} $$ is also a polynomial over $k$. This means that $p$ also divides $f$.