Prove that if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then $f(x)$ divides $s(x)g(x) +t(x)h(x)$.

Question

Let $F$ be a field. Prove that for all polynomials $f(x), g(x), h(x) \in {F}[x]$, if $f(x)$ divides $g(x)$ and $f(x)$ divides $h(x)$, then for all polynomials $s(x), t(x)\in {F}[x]$, $f(x)$ divides $s(x)g(x) +t(x)h(x)$.

How do I prove this question? I know that $f(x)=g(x)q(x)+r(x)$ but I'm not sure if I use that at all in this question.

Answer 1

Hint: How would you prove an equivalent statment for the integers?

Answer 2

If $f(x)$ divides $g(x)$, then $g(x) = f(x)q(x)$ and likewise $h(x) = f(x)r(x)$.

Also note that $f(x)$ divides $s(x)g(x)$, since $s(x)g(x) = s(x) f(x)q(x)$.