# Polynomial Inequalities

**Algebra**Level 5

Suppose \(f(x)\) and \(g(x)\) are non-constant polynomials with integer coefficients, such that \(f\) is monic and \[f(g(x))=(f(x))^2\cdot g(x).\] Suppose \(N\) is the number of possible polynomials \(g,\) such that all coefficients of \(g\) have absolute value strictly less than \(10^{100}\). Find the last three digits of \(N.\)

**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.