Abstract
The concepts of solvability in polynomials is highly imperative, most especially the concerned operations on the sums and the products of the roots. In this paper, an analysis of the fundamentals and the basic characterizations of the roots of polynomials was considered. This involves but is not limited to the sums as well as the products of their roots, not only those of the elementary or lower degrees but also that of any higher degree . Efforts have been intensified to state and prove certain characterizations which each case of the degrees of the polynomial must satisfy. Hence, the analysis further helps in determining of zeros for any given polynomial, including those of higher degrees.
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