  In this work, Some new inequalities for the numerical radius of block
$n$-by-$n$ matrices are presented. As an application, bounding of zeros of
polynomials using the Frobenius companion matrix partitioned by the Cartesian
decomposition approach is proved and affirmed by a numerical example showing
that our approach of bounding zeros of polynomials could be very effective in
comparison with the most famous and some recent results presented in the field.
