Proof by induction that $P_n(a) \neq 0$ for $n>3$.

Question

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?

Answer 1

If you expand that a bit, and use the binomial theorem:

$$P_n(a) = a^n + (a + c)^n - (a + b)^n $$

Now, $P_3(a) \ne 0 $, so:

$$a^3 + (a + c)^3 - (a + b)^3 \ne 0$$

Assume that $\forall \ 3 \le k \le n-1$, $P_k(a) \ne 0$.

What happens if $P_n(a) = 0$?

Answer 2

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