Proof by induction that $P_n(a) \neq 0$ for $n>3$. Let $a,b,c$ be 3 non-zero coprime integers and $P_n(a)=a^n+\sum_{k=1}^{n}{{n\choose{k}}a^{n-k}(c^k-b^k)}$
Show that if $P_3(a) \neq 0$ then for all $n \geq 3, P_n(a)\neq 0$
Using mathematical induction, how can I prove it?

Hint: $\sum_{k=1}^{n}{{n\choose{k}}a^{n-k}(c^k-b^k)} = \sum_{k=1}^{n}{{n\choose{k}}a^{n-k}c^k}-\sum_{k=1}^{n}{{n\choose{k}}a^{n-k}b^k}=(a+c)^n-(a+b)^n$
