# Remainder Theorem

How to solve the remainder theorem problems: theorem, examples, and their solutions.

## Theorem

Let's think of *f*(*x*) ÷ (*x* - *a*).

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

If *x* = *a*,*f*(*a*) = (*a* - *a*)⋅(quotient) + (remainder)

= (remainder).

So *f*(*a*) is the remainder.

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

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

This is the remainder theorem.

## Example 1

The zero of the divisor (*x* - 2) is 2.

So the remainder is *f*(2).

## Example 2

The zero of the divisor (*x* + 1) is -1.

So the remainder is *f*(-1).

## Example 3

The zero of the divisor (*x* - 1) is 1.

So the remainder is *f*(1).

*f*(1) = -3 + *a*

And it says the remainder is 11.

So *f*(1) = -3 + *a* = 11.