# Fancy Functions

Algebra Level 5

For how many positive integers $$b<1000$$ do there exist integers $$a$$ and $$c$$ and non-constant real polynomials $$f(x)$$ and $$g(x)$$ such that $$(f(x))^3+af(x)+b=(g(x))^3+c(g(x))^2$$ for all $$x?$$

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