Matrix roots of matrix polynomials and block-eigenvalues of matrices partitioned into blocks are considered in this paper. We address the questions of root-finding, root-localization and root-separation for polynomials over matrices by using matricial norms of matrices and matrix norms of matrices. Block-versions of Cayley-Hamilton Theorem and Leverrier-Faddeev Algorithm and, as well, the Kronecker Sum of matrices partitioned into blocks assist in building a matrix polynomial (\( \lambda \)-matrix) whose eigenvalues are the differences of the eigenvalues of a given matrix polynomial (\( \lambda \)-matrix). Some numerical examples are presented in the context of second-degree polynomials over commutative matrices.