# Theorem

The Remainder Theorem states that the remainder when the polynomial $P(x)$ is divided by $x-a$ (usually with synthetic division) is equal to the simplified value of $P(a)$

# Examples

## Example 1

What is the remainder in $\frac{x^2+2x+3}{x+1}$?

## Solution

Using synthetic or long division we obtain the quotient $x+1+\frac{2}{x^2+2x+3}$. In this case the remainder is $2$. However, we could've figured that out by evaluating $P(-1)$. Remember, we want the divisor in the form of $x-a$. $x+1=x-(-1)$ so $a=-1$.

$P(-1) = (-1)^2+2(-1)+3 = 1-2+3 = \boxed{2}$