Problem
ALG-B3-M08-P011 Polynomial Idempotent
#11
★★★★☆ Level 4 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(P(x))=P(x)\).
If \(P\) is nonconstant, its image is infinite.
Every constant polynomial works. Let \(P\) be nonconstant. Then the image of \(P\) is infinite. For every \(t=P(x)\), we have \(P(t)=t\). Thus the polynomial \(P(T)-T\) has infinitely many roots, so \(P(T)=T\). The answer is all constant polynomials and \(P(x)=x\).
Idempotence through an infinite image.