An irreducibility criterion for integer polynomials

Let $f(x)={\sum }_{i=0}^n{a}_i{x}^i$ be a polynomial with coefficients from the ring $\mathbb{Z}$ of integers satisfying either $(i)$ $0 < a_0 \leq a_{1} \leq \cdots \leq a_{k-1} < a_{k} < a_{k+1} \leq \cdots \leq a_n$forsome$k$,$0 \leq k \leq n-1$;or$(ii)$ $|a_n| > |a_{n-1}| + \cdots + |a_{0}|$with$a_0 \neq 0$\mathrm{.}Inthispaper,itisprovedthatif$|a_n|$or$|f(m)|$isaprimenumberforsomeinteger$m$with$|m|\geq 2 $ then the polynomial $f(x)$ is irreducible over $\mathbb{Z}$.