Problem
ALG-B3-M07-P004 Constant Difference
#4
★★☆☆☆ Level 2 of 5
Find all \(P\in\mathbb R[x]\) such that \(P(x+1)-P(x)=5\).
Compare with \(5x\).
The polynomial \(Q(x)=P(x)-5x\) satisfies \(Q(x+1)=Q(x)\). A periodic polynomial is constant. Hence \(Q(x)=c\), and \(P(x)=5x+c\). All such polynomials work.
Basic finite difference.