# What is the difference between the remainder theorem and the factor theorem?

##### 1 Answer

The two theorems are similar, but refer to different things.

See explanation.

#### Explanation:

The **remainder theorem** tells us that for any polynomial

The **factor theorem** tells us that if

For example, let's consider the polynomial

**Using the remainder theorem**

We can plug in

Therefore, by the remainder theorem, the remainder when you divide

You can also apply this in reverse. Divide

**Using the factor theorem**

The quadratic polynomial

This tells us that

We can also apply the factor theorem in reverse:

We can factor

Basically, the remainder theorem links the remainder of division by a binomial with the value of a function at a point, while the factor theorem links the factors of a polynomial to its zeros.