Final answer:
The Remainder Theorem states the remainder of a polynomial divided by (x - c) is f(c), and synthetic division is a quick method to divide polynomials that also gives the remainder, showing a practical application of the theorem.
Step-by-step explanation:
The Remainder Theorem and synthetic division are closely related concepts in polynomial algebra. The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder is f(c). Synthetic division is a method used to divide polynomials that provides a shorthand way of performing long division. It not only helps in finding the quotient but also directly gives the remainder when dividing by a linear term.
To apply synthetic division, we take the opposite of the constant term in the divisor, which would represent c in (x - c), and use it to perform the synthetic division process. The last number in the synthetic division process represents the remainder, which, according to the Remainder Theorem, should be the same as f(c). Thus, synthetic division is a practical application of the Remainder Theorem, especially when trying to quickly evaluate polynomials at certain points or when factoring polynomials.