Rotating the roots of a cubic

Algebra Level 5

Consider all pairs of polynomials (f(x),g(x)) (f(x), g(x)) with complex coefficients such that

  1. f(x)f(x) is a monic polynomial of degree 3 and has distinct roots α1,α2,α3 \alpha_1, \alpha_2, \alpha_3.
  2. g(x)g(x) is a linear polynomial such that g(α1)=α2, g(α2)=α3, g(α3)=α1.g(\alpha_1)=\alpha_2,\ g(\alpha_2)=\alpha_3,\ g(\alpha_3)=\alpha_1.
  3. f(0)=2f(0) = 2 .
  4. f(1)=9f(1) = 9 .

The sum of all possible (distinct) values of f(2)f(2) is NN. What are the last three digits of NN?

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x5 x^3 + 3x - 5 is monic but the polynomial x4+2x36 -x^4 + 2x^3 - 6 is not.


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