Exercise 27, § 1

Let f and g be two coprime polynomials in $K[X]$; show that every polynomial h such that $\deg(h) < \deg(fg)$ can be expressed uniquely in the form $h = uf + vg$ where u and v are polynomials such that $\deg(u) < \deg(g)$ and $\deg(v) < \deg(f)$. First give a proof analogous to that of Ex. 26 (based on the Euclidean algorithm), then give a proof in which the problem is considered as a linear problem.