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 when $x^2+2x+3$ is divided by $x+1$ ?

Solution

Using synthetic or long division we obtain the quotient $1+\frac{2}{x^2+2x+3}$ . 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 $a=-1$ .

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

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Remainder Theorem

Contents

Theorem

Examples

Example 1

Solution