On an irreducibility theorem of A. Schinzel associated with coverings of the integers

Let f(x) and g(x) be two relatively prime polynomials having integer coefficients with g(0) ≠ 0. The authors show that there is an N = N(f,g) such that if n ≥ N, then the non-reciprocal part of the polynomial f(x)xⁿ + g(x) is either irreducible or identically 1 or -1 with certain clear exceptions that arise from a theorem of Capelli. A version of this result is originally due to Andrzej Schinzel. The present paper gives a new approach that allows for an improved estimate on the value of N.