If a polynomial *P*( *x*) is divided by ( *x* – *r*), then the remainder of this division is the same as evaluating *P*( *r*), and evaluating *P*( *r*) for some polynomial *P*( *x*) is the same as finding the remainder of *P*( *x*) divided by ( *x* – *r*).

##### Example 1

Find *P*(–3) if *P*( *x*) = 7 *x* ^{5} – 4 *x* ^{3} + 2 *x* –11.

There are two methods of finding *P*(–3).

**Method 1:** Directly replace –3 for *x.*

**Method 2:** Find the remainder of *P*( *x*) divided by [ *x* – (–3)].

To use Method 1:

By Method 2, *P*( *x*) divided by [ *x* – (–3)] will be done by synthetic division.

Therefore, *P*(–3) = –1610.

##### Example 2

Find the remainder of *P*( *x*) divided by ( *x* – 4) if

*P*( *x*) = *x* ^{4} + *x* ^{3} – 13 *x* ^{2} – 25 *x* – 12

By Method 1,

By Method 2, synthetic division,

Therefore, the remainder = 0.

In Example
**divisor** (the number doing the dividing) and the **quotient** (the answer) are **factors** of the **dividend** (the expression being divided).