The theorem which I am referring to states: for any $f, g$ there exist $q, r$ such that $f(x)=g(x)q(x)+r(x)$ with the degree of $r$ less than the degree of $g$ if $g$ is monic.
The book I am using remarks that it can be proven via induction on the degree of $g$, but leaves the proof to the reader. Unfortunately, this reader is not clever enough to get it.
The base case is fairly clear, but I'm completely stuck after that. Any hints?