Problem
ALG-B3-M07-P002 Midpoint Polynomial
#2
★★☆☆☆ Level 2 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x+y)+P(x-y)=2P(x)\).
Check a polynomial of degree \(d\ge2\).
Let \(P\) have degree \(d\ge2\) and leading coefficient \(a\). The leading part on the left is \(a((x+y)^d+(x-y)^d)\), while on the right it is \(2ax^d\). Expanding produces a term with \(y^2x^{d-2}\) when \(d\ge2\), impossible. Hence \(d\le1\). All linear polynomials \(P(x)=ax+b\) work.
Jensen-type equation without continuity because the function is already a polynomial.