Nice to square you
Suppose \(f(x)\) is a monic integer polynomial of degree three. Find the largest possible number of distinct integers \(n\) such that \(f(n^2)=f(n)\).
Details and assumptions
A polynomial is monic if its leading coefficient is 1. For example, the polynomial \( x^3 + 3x - 5 \) is monic but the polynomial \( -x^4 + 2x^3 - 6 \) is not.