Problem
ALG-B3-M11-P004 Cubic Difference
#4
★★★★☆ Level 4 of 5
Find \(P\in\mathbb R[x]\) if \(P(n+1)-P(n)=3n^2+3n+1\) for all integers \(n\).
The right-hand side is \((n+1)^3-n^3\).
The polynomial \(P(x+1)-P(x)-((x+1)^3-x^3)\) has infinitely many roots, so it is identically zero. Thus \(P(x)-x^3\) is periodic with period \(1\), hence constant. The answer is \(P(x)=x^3+c\).
Polynomial method in a mixed set.