Fancy Functions

Algebra Level 5

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

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