3. Division of Algebraic Expressions

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

Example 1

On this page

Simplify \large{\frac{3ab(4a^2b^5)}{8a^2b^3}}

Example 2

Simplify \large{\frac{12m^2n^3}{(6m^4n^5)^2}}

Example 3

Simplify \large{\frac{6p^3q^2-10p^2q}{4q}}

Dividing by a Fraction

Recall the following when dividing algebraic expressions.

The reciprocal of a number x, is math formula.

For example, the reciprocal of 5 is math formula and the reciprocal of math formula is math formula.

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

For example, math formula

Example 4

Simplify

3 plus 1 on x

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:

question

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.

long division

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

long division

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.

LIVEMath

Example 7

Simplify math formula

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

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