On the factorization of the trinomials \(x^n + cx^{n-1} + d\)
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.
scientific article; zbMATH DE number 6078186