The Bernoulli Polynomials are a sequence of polynomials that are important in number theory and mathematical analysis. They are defined using the Bernoulli numbers, which are a set of rational numbers that arise in the study of series and number theory. The polynomials are typically denoted as B_n(x) , where n indicates the degree of the polynomial. These polynomials have various applications, including in the computation of sums of powers of integers and in the theory of Fourier series. They also play a role in numerical analysis and combinatorics, making them a valuable tool in both pure and applied mathematics.