Suppose $f(X) = (X − α)^r\cdot g(X)$, where $α ∈ \Bbb C$ is nonzero, $r ∈ \Bbb Z^+$, and $g ∈ \Bbb C[X]$ is nonzero. Prove that $||g|| < (1 + \deg g)\cdot (2 \max(1, |α|^{−1}))\cdot \deg f\cdot ||f||$
I wanted to try Newton's theory but failed. I am considering using substitution.