Problem
ALG-B3-M07-P016 Difference on Integers
#16
★★★★★ Level 5 of 5
Find all \(P\in\mathbb R[x]\) if \(P(n+1)-P(n)=n^2\) for all integers \(n\).
First turn the equality on integers into an identity.
The polynomial \(R(x)=P(x+1)-P(x)-x^2\) has infinitely many roots, so \(R\equiv0\). Thus the problem reduces to \(P(x+1)-P(x)=x^2\). As before, \(P(x)=\frac{x(x-1)(2x-1)}{6}+c\).
Important transition from infinitely many integer points to an identity.