0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ $f(x)$ 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 $4\cdot s + 20 \cdot t$? $\endgroup$ – Andrew Tawfeek Apr 2 at 1:49
1
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Would I use DIC for polynomials? $\endgroup$ – Sania Apr 2 at 1:40
  • $\begingroup$ What is DIC?... $\endgroup$ – Maria Mazur Apr 2 at 1:41
  • $\begingroup$ If a divides b and a divides c then for integers x,y a divides bx+cy $\endgroup$ – Sania Apr 2 at 1:41
  • $\begingroup$ Actualy this is an equivalent statement, you need to remember how do you prove it. $\endgroup$ – Maria Mazur Apr 2 at 1:43
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.