In this thesis, we apply numerical radius and spectral radius estimates to the Frobenius companion matrices of monic polynomials to derive new bounds for their zeros. We first give an introduction in Chapter 1. In Chapter 2, for a polynomial p, we let the Frobenius companion matrix of p be a 2 × 2 block matrix, i.e. C(p) = A B C D . We obtain estimates for the numerical radius of C(p) and derive new bounds for the zeros of polynomials. Some numerical examples are given to show that, depending on the polynomial p, our estimation is better than results established recently. In Chapter 3, we consider the Frobenius companion matrix is the sum of two matrices, i.e. C(p) = A + B. We find the numerical radius and the spectral radius of C(p) and give some improvements of bounds for zeros of a polynomial. Finally, some examples are shown to verify our improvements.