If f(x) and g(x) are relatively prime polynomials in ℤ[x] satisfying certain conditions arising from a theorem of Capelli and if n is an integer > N for some sufficiently large N, then the non-reciprocal part of f(x)xⁿ + g(x) is either identically ±1 or is irreducible over the rationals. This result follows from work of Schinzel in 1965. We show here that under the conditions that f(x) and g(x) are relatively prime 0,1-polynomials (so each coefficient is either 0 or 1) and f(0) = g(0) = 1, one can take N = deg g + 2max{deg f, deg g}.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-1-1,
author = {Michael Filaseta and Manton Matthews, Jr.},
title = {On the irreducibility of 0,1-polynomials of the form f(x)xn + g(x)},
journal = {Colloquium Mathematicae},
volume = {100},
year = {2004},
pages = {1-5},
zbl = {1060.11066},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-1-1}
}
Michael Filaseta; Manton Matthews, Jr. On the irreducibility of 0,1-polynomials of the form f(x)xⁿ + g(x). Colloquium Mathematicae, Tome 100 (2004) pp. 1-5. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm99-1-1/