Problem
ALG-B3-M07-P005 Linear Difference
#5
★★☆☆☆ Level 2 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x+1)-P(x)=2x+1\).
Subtract \(x^2\).
Since \((x+1)^2-x^2=2x+1\), the polynomial \(Q(x)=P(x)-x^2\) is periodic with period \(1\). Therefore \(Q\) is constant. The answer is \(P(x)=x^2+c\).
Preparation for harder finite differences.