# Remainder Theorem

How to use the remainder theorem to find the remainder of the division of a polynomial: theorem, 2 examples, and their solutions.

## Theorem

The remainder of f(x)/(x - a) is

f(a).

This is the remainder theorem.

## Example(x^{3} - 7x + 11)/(x - 2)

Set f(x) = x^{3} - 7x + 11.

Then (given) = f(x)/(x - 2).

The zero of (x - 2) is 2.

Then, by the remainder theorem,

the remainder of f(x)/(x - 2) is

f(2).

f(x) = x^{3} - 7x + 11

Then

f(2) = 2^{3} - 7⋅2 + 11.

2^{3} = 8

-7⋅2 = -14

8 - 14 = -6

-6 + 11 = 5

So the remainder is 5.

## Example(2x^{4} + x^{3} - 5x^{2} + 3x + 4) ÷ (x + 1)

Set f(x) = 2x^{4} + x^{3} - 5x^{2} + 3x + 4.

Then (given) = f(x) ÷ (x + 1).

The zero of (x + 1) is -1.

Then, by the remainder theorem,

the remainder of f(x) ÷ (x + 1) is

f(-1).

f(x) = 2x^{4} + x^{3} - 5x^{2} + 3x + 4

Then

f(-1) = 2⋅(-1)^{4} + (-1)^{3} - 5⋅(-1)^{2} + 3⋅(-1) + 4.

(-1)^{4} = 1

+(-1)^{3} = +(-1) = -1

(-1)^{2} = 1

+3⋅(-1) = -3

2⋅1 = 2

-5⋅1 = -5

-1 - 3 + 4 = 0

2 - 5 = -3

So the remainder is -3.