Which Cubics are Irreducible?

Algebra Level 5

For how many positive integers \(1 \leq k \leq 1000\) is the polynomial \(f_k(x)=x^3+x+k\) irreducible?

Details and assumptions

A polynomial with integer coefficients is called reducible if it equals to a product of two non-constant polynomials with integer coefficients. Otherwise, it is called irreducible.

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