Factoring The Difference of 2 Squares

After studying this lesson, you will be able to:

• Factor the difference of two squares.

Steps of Factoring:

1. Factor out the GCF

2. Look at the number of terms:

• 2 Terms: Look for the Difference of 2 Squares
• 3 Terms: Factor the Trinomial
• 4 Terms: Factor by Grouping

3. Factor Completely

4. Check by Multiplying

This lesson will concentrate on the second step of factoring: Factoring the Difference of 2 Squares.

**When there are 2 terms, we look for the difference of 2 squares. Don't forget to look for a GCF first.**

We have the difference of two squares when the following are true:

There are 2 terms separated by a minus sign

To factor the difference of 2 squares, we write 2 parentheses. One will have an addition sign and the other will have a subtraction sign like this:

Next, we find the square root of the first term. We put these in the first positions. Then, we find the square root of the constant term and we put these in the last positions.

Example 1

Factor

There is no GCF other than one. This is the difference of two squares. Now we take the square root of the first term. The square root of ¼ x ² is so we put in the first positions:

Now we take the square root of the constant term. The square root of is y so we put in the last positions. Now, the problem is completely factored.

Check by using FOIL

Example 2

Factor 12x ³ - 27xy ²

There is a GCF in this problem. Therefore, we have to factor out the GCF first. The GCF is 3x so we factor that out.

3x (4x ² - 9y ²)

Now we have the difference of two squares remaining in the parentheses. We have to factor completely so we factor the difference of two squares.....keeping the GCF.

3x (2x + 3y ) (2x - 3y )

Check (it will take two steps to check) First forget about the 3x for the time being and FOIL the 2 binomials:

(2x + 3y) (2x - 3y) 4x ² - 6xy + 6xy - 9y ² which is 4x ² - 9y ²

Now multiply the result by 3x:

3x (4x ² - 9y ²) 12x ³ - 27xy ²

Example 3

Factor 162x ⁴ - 32y⁸

There is a GCF in this problem. Therefore, we have to factor out the GCF first. The GCF is 2 so we factor that out.

2(81x ⁴ - 16y⁸ )

Now we have the difference of two squares remaining in the parentheses. We have to factor completely so we factor the difference of two squares.....keeping the GCF.

2(9x ² + 4y⁴ ) (9x ² - 4y⁴ )

The second parenthesis still contains the difference of 2 squares so we have to factor that again. We keep the GCF and the first parenthesis.

2(9x ² + 4y⁴ ) (3x + 2y ) (3x - 2y )

This is now factored completely.

Check (it will take three steps to check) First forget about the 2 and the first binomial for the time being and FOIL the last 2 binomials:

(3x + 2y ) (3x - 2y ) (9x ² - 4y⁴ )

Now multiply the 2 remaining binomials:

(9x ² + 4y⁴ ) (9x ² - 4y⁴ ) (81x ⁴ - 16y⁸ )

Now multiply the result by 2:

2(81x ⁴ - 16y⁸ ) 162x ⁴ - 32y⁸