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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)`


First, we multiply out the top line:


When we write it out in full, this means

`(12 xx aaa xx b b b b b b)/(8 xx aa xx b b b)`

Next, cancel the numbers top and bottom (we divide top and bottom by `4`), the "a" terms (we cancel `a^2=aa` from top and bottom) and the "b" terms (we cancel `b^3=b b b` from top and bottom) to give us the final answer:


Example 2

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


We square the denominator (bottom) of the fraction:


Next, we cancel out the numbers, and the "m" and "n" terms to give the final answer:


Example 3

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


With this example, we'll break it into 2 fractions, both with denominator 4q to make it easier to see what to do.


Next, we cancel the numbers and variables:


Finally, we combine the fractions:


After you have had some practice with these, you'll be able to do it without separating them into 2 fractions first.

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




I'll show you how to do this two different ways. It is worth seeing both, because they are both useful. You can decide which is easier ;-)

Solution 1 - Multiplying by the Reciprocal

I take the top expression (numerator) and turn it into a single fraction with denominator x.


We do likewise with the bottom expression (denominator):


So the question has become:


We think of the right side as a division of the top by the bottom:


To divide by a fraction, you multiply by the reciprocal:


The x's cancelled out, and we have our final answer, which cannot be simplified any more.

Solution 2 - Multiplying Top and Bottom

I recognise that I have "/x" in both the numerator and denominator. So if I just multiply top and bottom by x, it will simplify everything by removing the fractions on top and bottom.


I am really just multiplying by "1" and not changing the original value of the fraction - just changing its form.

So I multiply each element of the top by x and each element of the bottom by x and I get:


I cannot simplify any further.

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.

long division

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)`


We are dividing a polynomial of degree 2 by a polynomial of degree 1. This is algebraic long division.

Step 1: We look at the first term of (3x2 − 11x − 4) and the first term of (x − 4).

Divide as follows: 3x2 ÷ x = 3x

We write 3x at top of our long division and multiply (3x)(x − 4) = 3x2 − 12x to give the second row of our solution.

Division of polynomials

Step 2: Subtracting the second row from the first gives:

Division of polynomials

Be careful with

-11x − (-12x) = -11x + 12x = x

Step 3: Bring down the -4 from the first row:

Division of polynomials

Step 4: Divide x (in the 3rd row) by x from the (x − 4) in the question. Our answer is 1 and we write "+ 1" at the top of our long division.

Next, multiply (1) by (x − 4) to get the 4th row.

Division of polynomials

Step 5: Subtract the 4th row from the 3rd:

Division of polynomials

So (3x2 − 11x − 4) ÷ (x − 4) = 3x + 1

You can check your answer by multiplying (3x + 1) by (x − 4) and you'll get (3x2 − 11x − 4).

Example 7

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


We can think of `(6x^2+6+7x)/(2x+1)` as (6x2 + 7x + 6) ÷ (2x + 1)

Once again we are dividing a polynomial of degree 2 by a polynomial of lower degree (1). This is algebraic long division.

Step 1: 6x2 ÷ 2x = 3x

So we write the following, using (3x)(2x + 1) = 6x2 + 3x for the second row:

Division of polynomials

Step 2: We subtract 6x2 + 3x from the first row:

Division of polynomials

Step 3: Bring down the 6:

Division of polynomials

Step 4: Divide 4x by 2x. Our answer is 2 and we multiply 2(2x + 1) to get the 4th row.

Division of polynomials

Step 5: Subtract, and we are left with 4.

Division of polynomials

So the answer is:


NOTE: Some people prefer to write the problem with all the x2's, x's and units in line, as follows:

Division of polynomials

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

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