3. Division of Algebraic Expressions
Our first examples of division of algebraic expressions involve simplifying and canceling.
Example 1
On this page
Simplify 
Example 2
Simplify 
Example 3
Simplify 
Dividing by a Fraction
Recall the following when dividing algebraic expressions.
The reciprocal of a number x, is 
.
For example, the reciprocal of 5 is 
and the reciprocal of 
is 
.
To divide by a fraction, you multiply by the reciprocal of the fraction.
For example, 
Example 4
Simplify
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:
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:
We use a similar technique for long division in algebra.
Example 6 - Algebraic Long Division
Simplify (3x2 - 11x − 4) ÷ (x− 4)
Here is a LiveMath document which does it in one step. You can change things to see the effect. Mostly you will get a remainder of some sort, just like you do when dividing numbers.
Example 7
Simplify 
You can see how algebraic long division is used in a later section, Remainder and Factor Theorems.
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