# Remainder Theorem

How to use the remainder theorem to find the remainder of a rational expression: theorem, examples, and their solutions.

## Theorem

The remainder of *f*(*x*) ÷ (*x* - *a*) is*f*(*a*).

This is the remainder theorem.*f*(*x*) = (*x* - *a*)⋅(quotient) + (remainder)

So *f*(*x*) = (*x* - *a*)⋅(quotient) + *f*(*a*).

## Example 1: Remainder of (*x*^{3} - 7*x* + 11)/(*x* - 2)

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

Then the given expression is*f*(*x*)/(*x* - 2).

By the remainder theorem,

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

So (remainder) = *f*(2).

Put 2 into *f*(*x*).

Then *f*(2) = 2^{3} - 7⋅2 + 11.

2^{3} = 8

-7⋅2 = -14

-14 + 11 = -3

8 - 3 = 5

So the remainder of the given expression is 5.

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

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

Then the given expression is*f*(*x*)/(*x* - (-1)).

By the remainder theorem,

the remainder of *f*(*x*)/(*x* - (-1)) is*f*(-1).

So (remainder) = *f*(-1).

Put -1 into *f*(*x*).

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

2⋅(-1)^{4} = 2⋅1

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

-5⋅(-1)^{2} = -5⋅1

+3⋅(-1) = -3

2⋅1 = 2

-5⋅1 = -5

Cancel -1 and +1.

(dark gray terms)

Then the right side is, 2 - 5, -3.

So the remainder of the given expression is -3.

## Example 3: (*x*^{5} - 8*x*^{4} + 7*x*^{2} - 3*x* + *a*) ÷ (*x* - 1), Remainder 11, *a* = ?

Set *f*(*x*) = *x*^{5} - 8*x*^{4} + 7*x*^{2} - 3*x* + *a*.

Then the given expression is*f*(*x*) ÷ (*x* - 1).

By the remainder theorem,

the remainder of *f*(*x*) ÷ (*x* - 1) is*f*(1).

So (remainder) = *f*(1).

Put 1 into *f*(*x*).

Then *f*(1) = 1^{5} - 8⋅1^{4} + 7⋅1^{2} - 3⋅1 + *a*.

1^{5} = 1

-8⋅1^{4} = -8

+7⋅1^{2} = +7

-3⋅1 = -3

1 - 8 = -7

Cancel -7 and +7.

(dark gray terms)

Then (right side) = -3 + *a*.

It says

the remainder of the given expression is 11.

So the right side is also 11.

So set -3 + *a* = 11

and find the value of *a*.

Move -3 to the right side.

Then *a* = 14.