Problem
ALG-B3-M07-P012 Square of the Argument
#12
★★★★☆ Level 4 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x)^2=P(x^2)\).
If there is a nonzero root, build infinitely many roots.
The constant solutions are \(0\), \(1\). Let \(P\) be nonconstant. If \(r\ne0\) is a root, then \(P(r^2)=P(r)^2=0\), and then \(P(r^{2^k})=0\) for infinitely many \(k\), impossible. Thus the only root is \(0\), and \(P(x)=ax^n\). Substitution gives \(a^2=a\), hence \(a=1\). The answer is \(0\), \(1\), and \(x^n\), \(n\ge1\).
Root argument with a chain of powers.