Rotating the roots of a cubic

Algebra Level 5

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

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

The sum of all possible (distinct) values of \(f(2)\) is \(N\). What are 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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