Problem
ALG-B3-M08-P018 Iteration on the Image
#18
★★★★★ Level 5 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(P(x))=P(x)^2\).
If \(P\) is nonconstant, its image is infinite.
If \(P=c\) is constant, then \(c=c^2\), so \(c=0\) or \(c=1\). Let \(P\) be nonconstant. For every \(t\) in the image of \(P\), we have \(t=P(x)\), hence \(P(t)=t^2\). The image of a nonconstant polynomial is infinite, so \(P(T)-T^2\) has infinitely many roots. Therefore \(P(T)=T^2\). The answer is \(P=0\), \(P=1\), \(P(x)=x^2\).
Strong task on infinite image.