Irreducibility of polynomials with a large gap

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of N > deg(c) + deg(d) such that the polynomial f_N(x) = x^N c(x^-1) + d(x) is irreducible over ℚ, where c, d ∈ℤ[x] are polynomials with nonzero constant terms and satisfying suitable conditions. As an application, we show that x^N - k x^2 + 1 is irreducible for all N ≥ 5 and k ∈{3, 4, …, 24}∖{9, 16}. We also give a complete description of the factorization of polynomials of the form x^N + k x^N-1± (l x + 1) with k, l ∈ℤ, k ≠ l.