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)).
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 it means we can write . But it means . Therefore, . It follows that where .