Euclidean Division: Uniqueness and Remainder

This lecture covers the concept of Euclidean division for polynomials, focusing on the uniqueness of the quotient and the remainder. Through examples, the instructor demonstrates how to divide polynomials and determine the remainder, emphasizing the importance of the degree of the divisor. The lecture also introduces the theorem stating the existence of unique polynomials D(X) and R(X) such that D(X) multiplied by the divisor Q(X) plus R(X) equals the dividend P(X). Additionally, it explores the significance of the remainder in polynomial division, showcasing how the remainder is always of a degree lower than the divisor, with practical applications illustrated through the evaluation of polynomials at specific values.