The Remainder Theorem is a fundamental concept in algebra that relates to polynomial division. It states that when a polynomial f(x) is divided by a linear divisor of the form (x – c), the remainder of this division is equal to f(c). In simpler terms, if you plug in the value ‘c’ into the polynomial, the result will give you the remainder.
For example, consider the polynomial f(x) = 2x^2 + 3x + 5 and let’s divide it by (x – 1). According to the Remainder Theorem, to find the remainder, we simply evaluate f(1):
f(1) = 2(1)^2 + 3(1) + 5 = 2 + 3 + 5 = 10.
Thus, when f(x) is divided by (x – 1), the remainder is 10.
This theorem is particularly useful because it allows us to quickly find remainders without performing the full division process, which can be time-consuming. It also helps in determining if (x – c) is a factor of the polynomial. If the remainder is zero, it means that (x – c) is indeed a factor.