Problem
ALG-B3-M07-P014 Antiperiod
#14
★★★★☆ Level 4 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x+1)=-P(x)\).
Apply the condition twice.
Applying the condition twice gives \(P(x+2)=-P(x+1)=P(x)\). Thus \(P\) is periodic with period \(2\), so it is constant. Then \(c=-c\), hence \(c=0\). The answer is \(P\equiv0\).
Short task on impossibility of a nonconstant periodic polynomial.