FACTORIAL POLYNOMIALS AND POWERS - Vyshnevetskiy, Oleksandr (Physical and Mathematical Education, 2025)
Problem formulation. The main task of modern higher education is to develop students' natural abilities and talents, form competencies, develop critical thinking, and create conditions for ensuring the harmonious development of students. Thus, the problem of forming a holistic system of theoretical knowledge and practical skills in various disciplines, which will allow students to use the acquired knowledge to solve current issues, arises. However, the amount of teaching time allocated to studying sections of higher mathematics is constantly decreasing. Therefore, the issue of teaching a standard set of theorems (and, in general, sections of mathematics) in a small number of teaching hours is becoming more relevant. In this case, it is advisable not to proceed to a simple list of theorem formulations but to preserve the teaching of their proofs, which should be short. The purpose of the article is a concise presentation of one of the topics of the higher mathematics course, namely, the presentation of the topic "Newton's Interpolation Polynomial."Materials and Methods. The study used an analysis of scientific literature, particularly theorems of higher education institutions' traditional higher mathematics courses (section "Mathematical Analysis"), and the presentation of factorial polynomials and factorial powers.Results. The paper presents a derivation of Newton's interpolation formula for polynomials and (simultaneously) the properties of factorial polynomials, which is much shorter than in known textbooks. The proposed approach uses the properties of polynomials and powers. One of the generalizations of the factorial of a number is also presented, generalizing the concept of the binomial coefficient.Conclusions. Using the described approach, the teaching of the topic is significantly reduced, which can help study the specified topics in higher education institutions under conditions of limited study time.
Keywords: factorial polynomials; Newton formula; difference operator; shift operator; factorial powers