What is the Remainder Theorem

Learn about and how to apply the remainder and factor theorem. The remainder theorem states that f(a) is the remainder when the polynomial f(x) is divided by x - a. Thus, given a polynomial, f(x), which is to be divided by a linear binomial of the form x - a, the remainder of the division is given by f(a).

The factor theorem is a special case of the remainder theortem which states that if f(a) = 0, then x - a is a factor of the polynomial f(x). Thus, given a polynomial, f(x), to see if a linear binomial of the form x - a is a factor of the polynomial, we solve for f(a). If f(a) = 0, then x - a is a factor, and x - a is not a factor otherwise.

These two theorems help us understand how we better understand the relationship between a polynomial function, it's factors, and the remainder. Using the theorem also can save us time from determining if a we have a factor or zero of a polynomial without having to use division.
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