Intermediate Olympiad - Polynomials

Find all polynomials P(x) with the following property:

There exists a positive integer k such that:

P(P(x)) = [P(x)]^k


Good stuff. Anyway, you can easily discover some quick things by plugging in basic values of x.

Suppose r is a root of P(x) then, obviously, P(r) = 0

And, P(P(r)) = [P(r)]^k

P(P(0)) = 0^k = 0

P(0) = 0

Similarly, plugging in any root of P(x) yields that P(P(r)) = 0. Thus, all roots of P(x) are roots of P(P(x)). or:

P(P(r)) = P(r) = 0

P(P(r)) will equal 0 when P(r) is a root of P(x), or

P(r_1) = r_2

Since P(r_1) = o, we have:

r_2 = 0, for any root of P(x).

Thus, all roots of P(x) are 0., P(x) = cx^k for some k. Plugging this into our equation quickly yields c = 1, and thus, P(x) = x^k for all positive integers k.