# 3. Division of Algebraic Expressions

### Later, on this page

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

### Example 1

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

Answer

First, we multiply out the top line:

`(12a^3b^6)/(8a^2b^3)`

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:

`(3ab^3)/2`

### Example 2

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

Answer

We square the denominator (bottom) of the fraction:

`(12m^2n^3)/((6m^4n^5)^2)=(12m^2n^3)/(36m^8n^10)`

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

`1/(3m^6n^7)`

### Example 3

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

Answer

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

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

Next, we cancel the numbers and variables:

`(3p^3q)/2-(5p^2)/2`

Finally, we combine the fractions:

`(3p^3q-5p^2)/2`

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

Simplify

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

Answer

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*.

`3+1/x=(3x+1)/x`

We do likewise with the bottom expression (denominator):

`5/x+4=(5+4x)/x`

So the question has become:

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

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

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

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

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

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.

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

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:

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

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:

`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)`

Answer

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 (3*x*^{2} − 11*x* − 4) and the first term of (*x* − 4).

Divide as follows: 3*x*^{2} ÷ *x* = 3*x*

We write 3*x* at top of our long division and multiply (3*x*)(*x* − 4) = 3*x*^{2} − 12*x* to give the second row of our solution.

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

Be careful with

-11

x− (-12x) = -11x+ 12x=x

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

**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.

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

So (3*x*^{2} − 11*x* − 4) ÷ (*x* − 4) = 3*x* + 1

You can check your answer by multiplying (3*x* + 1) by (*x* − 4) and you'll get (3*x*^{2} − 11*x* − 4).

### Example 7

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

Answer

We can think of `(6x^2+6+7x)/(2x+1)` as (6*x*^{2} + 7*x* + 6) ÷ (2*x* + 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:** 6*x*^{2} ÷ 2*x* = 3*x*

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

**Step 2: **We subtract 6*x*^{2} + 3*x* from the first row:

**Step 3:** Bring down the 6:

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

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

So the answer is:

`(6x^2+6+7x)/(2x+1)=3x+2+4/(2x+1)`

**NOTE: **Some people prefer to write the problem with all the *x*^{2}'s, *x*'s and units in line, as follows:

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