Polynomials are fundamental mathematical expressions that play important roles in many fields of mathematics, including algebra, calculus, and geometry. They are useful for modeling real-world phenomena, solving equations, and analyzing data. The structure of a polynomial makes it easier to understand functions and their behavior, allowing for the development of mathematical models to describe numerous processes and relationships. Polynomials also serve an important role in calculus, providing foundations for more complicated mathematical ideas like differentiation and integration. Polynomials are fundamental to mathematical reasoning and problem-solving techniques because of their versatility and simplicity, which enhances our comprehension of the quantitative features of the world. This book will explore recent advancements in both analytical and numerical solutions of polynomials, examining their applications across scientific, engineering, and social domains. Furthermore, this book will thoroughly discuss and analyze the dynamical aspects of numerical techniques for solving polynomial equations. It will explore how these numerical methods interact with the dynamical behavior of polynomial systems, providing insights into convergence properties, stability, and efficiency. Through detailed analysis and case studies, readers will gain a deeper understanding of the relationship between numerical algorithms and the dynamical behavior of polynomial equations, enhancing their ability to apply these techniques effectively in various scientific and engineering contexts.