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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:

(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)

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)

With this example, we'll break it into 2 fractions, both with denominator 4q 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)

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)

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. Step 2: Subtracting the second row from the first gives: Be careful with

-11x − (-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 (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: Step 2: We subtract 6x2 + 3x from the first row: Step 3: Bring down the 6: Step 4: Divide 4x by 2x. Our answer is 2 and we multiply 2(2x + 1) to get the 4th row. Step 5: Subtract, and we are left with 4. (6x^2+6+7x)/(2x+1)=3x+2+4/(2x+1)

NOTE: Some people prefer to write the problem with all the x2'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.

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