We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > operatorname{deg}(c) + operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is irreducible over $mathbb Q$, where $c, d in mathbb Z[x]$ 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 $N ge 5$ and $k in {3, 4, ldots, 24} setminus {9, 16}$. We also give a complete description of the factorization of polynomials of the form $x^N + k x^{N-1} pm (l x + 1)$ with $k, l in mathbb Z$, $k eq l$.
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