Let F be a field and let e(x), g(x), h(x), and f(x) be polynomials in F[x] with h(x) of positive degree. Prove that if e(x) = gcd(g(x),h(x)) and e(x) divides f(x), then there is a polynomial j(x)in F[x] such that
g(x)j(x)= f(x)(modh(x)).
Since $\displaystyle e(x)=\gcd(g(x),h(x))$ it means we can write $\displaystyle e(x) = a(x)g(x)+b(x)h(x)$. But $\displaystyle e(x)|f(x)$ it means $\displaystyle f(x) = e(x)d(x)$. Therefore, $\displaystyle f(x) = e(x)d(x) = a(x)d(x)g(x)+b(x)d(x)h(x)$. It follows that $\displaystyle g(x)j(x) \equiv g(x) ~ \bmod h(x)$ where $\displaystyle j(x)=a(x)d(x)$.