On the factorization of the trinomials \(x^n + cx^{n-1} + d\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1293.12003
DOI: 10.1142/S179304211250090X
author: Joshua Harrington
published in: International Journal of Number Theory
publication date: 2012-09-06T00:00:00Z
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 6078186
zbMATH Keywords: Polynomials
zbMATH Keywords: irreducibility
zbMATH Keywords: trinomials
zbMATH Keywords: factorization
MaRDI profile type: Publication
cites work: On algebraic equations with all but one root in the interior of the unit circle. To my teacher and former colleague Erhard Schmidt on his 75th birthday
title: On the factorization of the trinomials \(x^n + cx^{n-1} + d\)
review text: The author is studying the irreducibility of \(f(x)=x^n+cx^{n-1}+\ldots+cx+c\in\mathbb{Z}[x].\) The main result is that if \(n\) and \(c\) are positive integers with \(c\geq 2,\) then the polynomials NEWLINE\[NEWLINEf(x)=x^n+\mathop\sum\limits_{j=0}^{n-1}cx^j, g(x)=x^n+\mathop\sum\limits_{j=0}^{n-1}(-1)^{n-j}cx^j, h(x)=x^n-\mathop\sum\limits_{j=0}^{n-1}cx^jNEWLINE\]NEWLINE and NEWLINE\[NEWLINEk(x)=x^n-\mathop\sum\limits_{j=0}^{n-1}(-1)^{n-j}cx^jNEWLINE\]NEWLINE are irreducible in \(\mathbb{Z}[x],\) with the exception of \(x^2\pm 4x+4.\) To prove this result a nice theorem about the factorization of trinomials of the form \(x^n\pm cx^{n-1}\pm d\) is proved. Some open questions are mentioned.
reviewed by: Cristodor-Paul Ionescu
description: scientific article; zbMATH DE number 6078186