For \(\ell=0,1,\ldots,m\), given \(n\times n\) complex matrices \(A_\ell\) and \(\Delta_\ell\), we can define the matrix polynomial \[ P(\lambda)=A_m\lambda^m+A_{m-1}\lambda^{m-1}+\cdots+A_1\lambda+A_0, \] and a perturbation of the matrix polynomial \(P(\lambda)\) of the form \[ P_\Delta(\lambda)=(A_m+\Delta_m)\lambda^m+(A_{m-1}+\Delta_{m-1})\lambda^{m-1}+ \cdots+(A_1+\Delta_1)\lambda+A_0+\Delta_0. \] For a given \(\varepsilon>0\) and a given set of nonnegative weights \(w=\{w_0,w_1,\ldots,w_m\}\), with at least one nonzero element, the \(\varepsilon\)-pseudospectrum of \(P(\lambda)\) is defined by \[ \sigma_{\varepsilon,w}=\{\lambda\in \mathbb{C}\, :\, \det P_\Delta(\lambda)=0, \| \Delta_\ell\| _2\leq \varepsilon w_\ell, \, \ell=0,1,\ldots,m\}. \] The author studies the \(\varepsilon\)-pseudospectrum of \(P(\lambda)\) constructing a lower bound for the distance between \(\sigma_{\varepsilon,w}\) and a given point out of \(\sigma_{\varepsilon,w}\).