Problem
ALG-B3-M07-P019 The Image as an Infinite Set
#19
★★★★★ Level 5 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(P(x))=P(x)^2\) for all \(x\).
If \(P\) is nonconstant, its image is infinite.
If \(P\) is constant, say \(P=c\), then \(c=c^2\), so \(c=0\) or \(c=1\). Now let \(P\) be nonconstant. Then the image of \(P\) is infinite. For every number \(t\) in this image, there is \(x\) such that \(t=P(x)\), and the condition gives \(P(t)=t^2\). Hence the polynomial \(P(T)-T^2\) has infinitely many roots, so it is identically zero. Therefore \(P(x)=x^2\). Checking shows that \(0\), \(1\), and \(x^2\) all work.
Strong task using the infinite image of a nonconstant polynomial.