An irreducibility criterion for polynomials over integers

In this article, we consider the polynomials of the form $f(x)=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n\in \mathbb{Z}[x],$ where $\mathrm{\mid }{a}_0\mathrm{\mid }=\mathrm{\mid }{a}_1\mathrm{\mid }+\cdots +\mathrm{\mid }{a}_n\mathrm{\mid }$ and $\mathrm{\mid }{a}_0\mathrm{\mid }$ is a prime. We show that these polynomials have a cyclotomic factor whenever reducible. As a consequence, we give a simple procedure for checking the irreducibility of trinomials of this form and separability criterion for certain quadrinomials.