Irreducibility and locus of complex roots of polynomials related to Fermat's Last Theorem

We investigate the polynomials xⁿ + (1-x)ⁿ + aⁿ, a rational root of which would provide a counterexample to Fermat's Last Theorem. We consider the more general question of their irreducibility and prove that in some cases. We investigate the location of complex roots of these polynomials, and prove that for some a ∈ Q, the roots lie on explicitly given curves while being dense in those curves.