Problem
ALG-B3-M07-P003 Quadratic Polynomial
#3
★★☆☆☆ Level 2 of 5
Find all \(P\in\mathbb R[x]\) if \(P(x+y)+P(x-y)=2P(x)+2P(y)\) and \(P(1)=2\).
Check that the degree cannot exceed \(2\).
Putting \(y=0\), we get \(P(0)=0\). Comparing leading parts shows that the degree is at most \(2\). Let \(P(x)=ax^2+bx\). Substitution gives \(2ax^2+2ay^2+2bx=2ax^2+2bx+2ay^2+2by\), hence \(b=0\). From \(P(1)=2\), \(a=2\). The answer is \(P(x)=2x^2\).
Quadratic version of Cauchy-type equations.