Problem
ALG-B3-M11-P014 Idempotent Polynomial
#14
★★★★★ Level 5 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(P(x))=P(x)\).
If \(P\) is nonconstant, its image is infinite.
Constant polynomials work. If \(P\) is nonconstant, then for infinitely many \(t=P(x)\), \(P(t)=t\). Hence \(P(T)-T\equiv0\). The answer is all constant polynomials and \(P(x)=x\).
Composition through image.