# 3. Division of Algebraic Expressions

Our first examples of division of algebraic expressions involve simplifying and canceling.

### Example 1

### On this page

Simplify `(3ab(4a^2b^5))/(8a^2b^3)`

### Example 2

Simplify `(12m^2n^3)/((6m^4n^5)^2)`

### Example 3

Simplify `(6p^3q^2-10p^2q)/(4q)`

## Dividing by a Fraction

Recall the following when dividing algebraic expressions.

The **reciprocal** of a number *x*, is `1/x`.

For example, the reciprocal of 5 is `1/5` and the reciprocal of `1 2/3` is `3/5`.

To **divide** by a fraction, you multiply by
the **reciprocal** of the fraction.

For example, `3/4 -: 7/x=3/4xxx/7=(3x)/28`

### Example 4

Simplify

`(3+1/x)/(5/x+4)`

## Long Division in Algebra

Before we do an example using algebra, let’s remember how to do long division with **numbers** first.

### Example 5

Let’s do 23,576 divided by 13.

We can write this as a fraction:

`23576/13`

Now, to divide this, (assuming we do not have a calculator) we could proceed as follows.

23 divided by 13 = 1 with remainder 10.

We bring the 5 (the next number after 3) down.

Now we have

105 divided by 13 is 8 with remainder 1

We continue until we get to the last number, 6.

Our result means that the answer is 1,813 with remainder 7, or:

`23576/13=1813 7/13`

We use a similar technique for long division in algebra.

### Example 6 - Algebraic Long Division

Simplify `(3x^2-11x-4)-:(x-4)`

### Example 7

Simplify `(6x^2+6+7x)/(2x+1)`

You can see how algebraic long division is used in a later section, Remainder and Factor Theorems.

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