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

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.

• $f(x)$ is a factor of both $g(x)$ and $h(x)$. Consider the case where $f(x)=2$, $g(x)=4$, and $h(x)=20$. Does $f(x)$ divide $4\cdot s + 20 \cdot t$? – Andrew Tawfeek Apr 2 at 1:49

## 2 Answers

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

• Would I use DIC for polynomials? – Sania Apr 2 at 1:40
• What is DIC?... – Maria Mazur Apr 2 at 1:41
• If a divides b and a divides c then for integers x,y a divides bx+cy – Sania Apr 2 at 1:41
• Actualy this is an equivalent statement, you need to remember how do you prove it. – Maria Mazur Apr 2 at 1:43

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 $$s(x)g(x) = s(x) f(x)q(x)$$.