Let's think of a division of natural numbers. Share **7 out of 5 **means get the quotient **1** and the rest **2**. We can write:

Now let's think about the division of the polynomial **A (x)** by the nonzero polynomial **B (x)**which generates the quotient **Q (x)** and the rest **R (x)**.

In this division:

**A (x)**it is the dividend;**B (x)**is the divisor;**Q (x)**is the quotient;**R (x)**is the rest of the division.

The degree of **R (x)** must be less than the degree of **B (x)** or **R (x) = 0**.

When **A (x)** is divisible by **B (x)**, we say the division is exact, that is, **R (x) = 0**.

## Example 1

Determine the quotient of :

**Resolution**

- We divide the highest degree dividend term by the highest divisor term. The result will be a quotient term:

- We multiply
**x²**per**B (x)**and subtract the product from**A (x)**, getting the first partial rest:

- We divide the highest degree term of the first partial remainder by the highest degree term of the divisor, and we get as a result a quotient term:

- We multiply
**-2x**per**B (x)**and subtract the product from the first partial remainder, obtaining the second partial remainder:

- We divide the highest degree term of the second partial remainder by the highest degree term of the divisor, and we get as a result a quotient term:

- We multiply
**1**per**B (x)**and subtract the product from the second partial remainder:

Since the degree of rest is less than the degree of divisor, the division is closed.

We verified that:

## Example 2

Determine the quotient of :

*Resolution*

We easily verify that:

In these two examples, we use the **key method** to effect the division of polynomials.

By the examples we find that:

**quotient degree** = **dividend degree **- **divider degree**