On the reducibility of certain quadrinomials
Jonas Jankauskas · Glasnik Matematicki · 2010
asked to determine all irreducible polynomials of the form P(x) = x i + x j + x k + 4 with integer exponents i> j> k> 0, such that for some positive integer l the polynomial P(x l) is reducible in Z[x]. In this paper we prove that such polynomials are quadrinomials x 4m + x 3m + x 2m + 4, where m is an odd positive integer. In addition, Walsh asked for the examples of reducible quadrinomials x i +x j +x k +n, n> 4 with no linear or quadratic factors. We compute the examples of reducible polynomials of the form above with non-trivial factors and negative coefficient n. 1.