Problem
ALG-B3-M07-P013 Cube of the Argument
#13
★★★★☆ Level 4 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x)^3=P(x^3)\).
Repeat the root argument and compare leading coefficients.
Constant solutions satisfy \(c^3=c\), so \(c=0,1,-1\). Let \(P\) be nonconstant. A nonzero root would generate an infinite chain of roots \(r,r^3,r^9,\ldots\), so the only root is \(0\). Hence \(P(x)=ax^n\). Substitution gives \(a^3=a\), so \(a=1\) or \(a=-1\). The answer is \(0\), \(1\), \(-1\), \(x^n\), and \(-x^n\), \(n\ge1\).
Differs from the square case by the sign of the leading coefficient.