The Factorial Phenomenon: Unraveling the Mysteries of Finite Differences in Polynomials

Abstract This paper investigates the use of finite differences for polynomial functions of degree n. Initially, we observed the only common number in the nth steps differences between each consecutive power of the nth exponent of the ordinal numbers is the factorial of n. After that, we demonstrated that the nth order finite differences of such functions are constant with a value equal to n!. Specifically, we generate a sequence of powers of the ordinary numbers with an exponent of n, compute the finite differences of this sequence for each order from 1 to n, and observe that the result is a constant value equal to n! for each order. We have provided a proof of this observation using mathematical induction and showed that it holds true for all positive integers. We have constructed a mathematical formula that supports our statement. It is, Δn f(x) = n! . We have also presented a pseudo-code that implement the concept of finite differences and demonstrate its application for different values of n. The insights we've gathered hold significant value for the examination and construction of polynomial functions, offering applicable knowledge across diverse disciplines such as mathematics, engineering, and computer science.