# 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.

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