P(x) is a monic trinomial of degree 2 such that P(x) and P(P(P(x))) share a root. Prove that P(0)*P(1)=0.
Let r be the root shared by p(x) and p(p(p(x))). Then p(r)=0 and p(p(p(r)))=0. Substitute the first into the second and get p(p(0))=0. This implies p(0) is a root of p(x).
p is given to be monic, thus p(0) is the product of both roots of p(x). There are only two ways for p(0) to be both a root and a product of two roots: p(0)=0 or the other root equals 1. In the case the other root equals 1 then p(1)=0.
So either p(0)=0 or p(1)=0. Then p(0)*p(1)=0 in either case. QED
Solution
