# 3. Division of Algebraic Expressions

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

### Example 1

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)

Continues below

## 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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