The Legendre Polynomials are a set of orthogonal polynomials that arise in solving problems in physics and engineering, particularly in the context of potential theory and quantum mechanics. They are defined on the interval from -1 to 1 and are denoted as P_n(x) , where n is a non-negative integer. These polynomials can be generated using Rodrigues' formula, which provides a way to express them mathematically. Legendre Polynomials have important properties, such as orthogonality, which means that the integral of the product of two different polynomials over the interval is zero. They also satisfy the Legendre differential equation, making them useful in various applications, including spherical harmonics and numerical analysis.