1. Factoring the Difference of
Algebra 1 ~ Chapter 9.5
Factoring the Difference of
Two Perfect Squares

2. A polynomial is a difference of two squares if:
There are two terms, one subtracted from the other.
Both terms are perfect squares.
4x2 – 9
2x 2x 

3. To Factor the difference of perfect squares, take the square root of each term. In one set of parenthesis the sign is positive, and in the other it is negative.

4. Recognize a difference of two squares:
-the coefficients of variable terms are perfect squares
-powers on variable terms are even
-constants are perfect squares.
Reading Math

5. Example 1: Recognizing and Factoring the Difference of Two Squares
Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
3p2 – 9q4
3p2 – 9q4
3q2  3q2
3p2 is not a perfect square.
3p2 – 9q4 is not the difference of two squares because 3p2 is not a perfect square.

6. Example 2: Recognizing and Factoring the Difference of Two Squares
Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
100x2 – 4y2
100x2 – 4y2
2y 2y 
10x 10x
The polynomial is a difference of two squares.
(10x + 2y)(10x – 2y)
Write the polynomial as (a + b)(a – b).
100x2 – 4y2 = (10x + 2y)(10x – 2y)

7. Example 3: Recognizing and Factoring the Difference of Two Squares
Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
x4 – 25y6
x4 – 25y6
5y3 5y3 
x2 x2
The polynomial is a difference of two squares.
Write the polynomial as (a + b)(a – b).
(x2 + 5y3)(x2 – 5y3)
x4 – 25y6 = (x2 + 5y3)(x2 – 5y3)

8. Check It Out! You try it…
Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
1 – 4x2
1 – 4x2
2x 2x 
(1 + 2x)(1 – 2x)
1 – 4x2 = (1 + 2x)(1 – 2x)

9. Check It Out! You try another one…
Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
p8 – 49q6
p8 – 49q6
7q3 7q3 ●
p4 p4
(p4 + 7q3)(p4 – 7q3)
p8 – 49q6 = (p4 + 7q3)(p4 – 7q3)

10. Check It Out! Try one more…
Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
16x2 – 4y5
16x2 – 4y5
4x  4x
16x2 – 4y5 is not the difference of two squares because 4y5 is not a perfect square.

11. Example 4: Combining with GCF
Factor the binomial completely.
48a3 – 12a
12a(4a2 – 1)
12a(2a -1)(2a +1)

12. Example 4b: You try one…
Factor the binomial completely.
3b3 – 27b

13. Example 5: Doing more than one technique
Factor the binomial completely.
2x4 – 162
2(x4 – 81)
2(x2 -9)(x2 +9)
2(x - 3)(x + 3)(x2 +9)

14. Example 5b: You try one…
Factor the binomial completely.
4y4 – 2500

15. Example 6: Combining Factoring by grouping
Factor the binomial completely.
5x3 +15x2 – 5x - 15

16. Example 6: Your turn…
Factor the binomial completely.
6x3 +30x2 – 24x - 120

17. Example 7: Solving Equations
Solve the following:
p2 – 25 = 0

18. Example 7b: Solving Equations
Solve the following:
18x3 = 50x

19. Example 7c: You try one…
Solve the following:
48y3 = 3y