# Remainder Theorem

## Theorem

The Remainder Theorem states that the remainder when the polynomial is divided by (usually with synthetic division) is equal to the simplified value of .

## Proof

Let , where is the polynomial, is the divisor, is the quotient, and is the remainder. This equation can be rewritten as If , then substituting for results in

## Extension

An extension of the Remainder Theorem could be used to find the remainder of a polynomial when it is divided by a non-linear polynomial. Note that if is a polynomial, is the quotient, is a divisor, and is the remainder, the polynomial can be written as Note that the degree of is less than the degree of . Let be a root of , where is an integer and . That means for all , Thus, the points are on the graph of the remainder. If all the roots of are unique, then a system of equations can be made to find the remainder .

## Examples

### Introductory

- What is the remainder when is divided by ?

*Solution*: Using synthetic or long division we obtain the quotient . In this case the remainder is . However, we could've figured that out by evaluating . Remember, we want the divisor in the form of . so .
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