An irreducible class of polynomials over integers

In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}{x}^{n_1}+a_{n_2}{x}^{n_2}+\cdots +a_{n_r}{x}^{n_r}\in \mathbb{Z}[x],$ where $\mathrm{\mid }{a}_0\mathrm{\mid }\ge \mathrm{\mid }{a}_{n_1}\mathrm{\mid }+\cdots +\mathrm{\mid }{a}_{n_r}\mathrm{\mid },$ $\mathrm{\mid }{a}_0\mathrm{\mid }$ is a prime power and $\mathrm{\mid }{a}_0\mathrm{\mid }\nmid \mathrm{\mid }{a}_{n_1}{a}_{n_r}\mathrm{\mid }$. We will show that under the strict inequality these polynomials are irreducible for certain values of $n_1$. In the case of equality, apart from its cyclotomic factors, they have exactly one irreducible non-reciprocal factor.