Taali's Formula: On the elegant Symmetric Duality in the nature of Polynomials and its roots. [PART-I]

This paper presents an organized compilation of my independent research notes exploring the deep structural relations between polynomial roots and derivatives. It uses shorthand cyclic notations thus the paper is not 100% rigorous since many results were derived from Trial-and-Error and verified by the Principle of Mathematical Induction. Rather than claiming novel mathematical ground, this work acknowledges its foundation in standard manipulations and potential rediscoveries. However, the primary value lies in the resulting final symmetry, which will thrill you! (I promise, Enjoy). Keywords: Vieta’s relations of polynomial root’s sums and products, logarithmic differentiation, Leibniz product rule, Taylor’s theorem and Maclaurin expansion, polynomial root distribution, recurrence relations, sequences and series, theory of equations, Newton’s Formula on the relation of polynomial roots, fundamental theorem of algebra, remainder theorem, factor theorem, Principle Of Mathematical Induction. Pre-University Algebra & Calculus with Olympiad Style manipulations.