## 1 Introduction  
The problem of finding the zeros of complex monic polynomials of the form  
\[P_{n}(z)=z^{n}+a_{n}z^{n-1}+\cdots+a_{2}z+a_{1},\qquad z\in\mathbb{C},n\geq 2,a_{1}\
eq 0,\]  
is one of the most interesting, difficult, and oldest problems in mathematics. Several approaches have been taken to study this problem, with significant effort expended. See, for example [1, 2], and the references therein.  
One of the most common problems in this topic is determining or approximating the radius of the disk containing these zeros. In modern mathematics, the numerical range of a given Hilbert space is a very important concept related to the problem of bounding zeros of complex polynomials. The numerical range in matrix analysis (finite dimensional Hilbert space) is equivalent to the Rayleigh quotient for a given complex Hermitian matrix \(C\). It can be easily shown that, for a given matrix \(C\), the Rayleigh quotient reaches its smallest and largest eigenvalues of \(C\). In terms of the field of values of a complex \(n\times n\) matrix, one of the well-known tools used in this problem is the Frobenius Companion matrix \(C(P_{n})\), which corresponds to the polynomial \(P_{n}\). Therefore, inequalities for the numerical radius of such matrices would be very helpful in this direction. See [3, 4].  
The spectrum of a non-singular matrix \(C\) is the set of all its eigenvalues. Because the spectral set is contained in the numerical range, the radius of the spectral set is less than (or equal to) the radius of the numerical range. The problem with dealing with a spectral radius is that it is not as easy to deal with as a numerical radius. Because of that, in the last two decades, the problem of bounding the numerical radius has caught the attention of many researchers. It is worth mentioning that when an operator (matrix) \(C\) is a normal operator, the spectral radius is the same as the numerical radius.  
We continue our investigation of the numerical radius of Hilbert space operators in this article by establishing new inequalities for \(n\times n\) operator matrices for general Hilbert space operators. Our approach depends mainly on using the Cartesian decomposition of a general Hilbert space operator. To the best of our knowledge, no known algorithm has been employed to constrain the zeros of the polynomial \(P_{n}(z)\) using the Frobenius companion matrix partitioned using the Cartesian decomposition method, and this is one of the main originalities of the study. In particular, we remark that applying the numerical radius inequalities to the companion matrix presents a partition difficulty because the primary diagonal in the reported results necessitates a square sub-matrix. To that end, we show that the standard approaches for splitting the companion matrix, considered by many authors in the literature, are ineffective. As applications, several bounds for the zeros of complex polynomials are introduced. Various numerical examples show that our results are far superior to most of the well-known bounds found in the literature.  
The rest of the article is structured as follows: Section 2 presents the mathematical background of the study. Section 3 describes the numerical radius inequalities of the matrix operator. Applications for bounding zeros of polynomials are given in Section 4. The numerical radius of the real and imaginary parts of the considered matrix is determined in Section 5. Observations and discussion of the study are provided in Section 6. Concluding remarks are given in Section 7.