Problem
ALG-B3-M07-P006 Quadratic Difference
#6
★★★☆☆ Level 3 of 5
Find all \(P\in\mathbb R[x]\) if \(P(x+1)-P(x)=x^2\).
Find a polynomial whose difference is \(x^2\).
The polynomial \(F(x)=\frac{x(x-1)(2x-1)}{6}\) has difference \(F(x+1)-F(x)=x^2\). Then \(Q=P-F\) is periodic with period \(1\), hence constant. The answer is \(P(x)=\frac{x(x-1)(2x-1)}{6}+c\).
A finite difference of degree \(2\) raises the degree to \(3\).