In this paper we determine, in the complex plane, regions containing the zeros of the polynomial $P(z)=z^n+a_1z^{n-1}+a_2z^{n-2}+ \dots +a_{n-1}z+a_n$, $n \geq 3$. We also obtain two expressions which represent upper bounds for the moduli of the zeros of $P(z)$ with greater precision than those obtained by Cauchy and P. Montel.