algebra precalculus - Proving no polynomial $P(x)$ exists such that $P(a) = b$, $P(b) = c$, $P(c) = a$

If $P(x)$ is a polynomial with integer coefficients and $a, b ,c$ are three distinct integers, then show that it is impossible to have $P(a) = b$, $P(b) = c$, $P(c) = a$.

Answer

Hint: $b-c$ is divisible by $a-b$, and ...