Legendre Polynomials are a set of orthogonal polynomials that arise in solving problems in physics and engineering, particularly in the context of spherical harmonics and potential theory. 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 in terms of derivatives. These polynomials have important properties, such as orthogonality, which means that the integral of the product of two different Legendre polynomials over the interval is zero. This property makes them useful in expanding functions in series, similar to Fourier series. Applications of Legendre polynomials include solving Laplace's equation in spherical coordinates and modeling physical phenomena like gravitational fields.