1. Factoring Special Products
7-5
Factoring Special Products
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

2. Warm Up
Determine whether the following are perfect squares. If so, find the square root. 64
2. 36
yes; 6
3. 45
4. x2
yes; x
5. y8

3. ESSENTIAL QUESTION
How do you factor perfect-square trinomials and the difference of two squares?

5. Example 1A: Recognizing and Factoring Perfect-Square Trinomials
Determine whether each trinomial is a perfect square. If so, factor. If not explain.
9x2 – 15x + 64
9x2 – 15x + 64
8 8
3x 3x
2(3x 8) 
2(3x 8) ≠ –15x. 
9x2 – 15x + 64 is not a perfect-square trinomial because –15x ≠ 2(3x  8).

6. Example
Determine whether each trinomial is a perfect square. If so, factor. If not explain.
Method 2 Use the rule.
81x x + 25
a = 9x, b = 5
(9x)2 + 2(9x)(5) + 52
Write the trinomial as a2 + 2ab + b2.
Write the trinomial as (a + b)2.
(9x + 5)2

7. Example: Recognizing as Perfect-Square Trinomials
36x2 – 10x + 14
36x2 – 10x + 14
36x2 – 10x + 14 is not a perfect-square trinomial.

8. Example
Determine whether each trinomial is a perfect square. If so, factor. If not explain.
Method 1 Factor.
x2 + 4x + 4
Factors of 4 Sum
(1 and 4) 5
(2 and 2) 4  
(x + 2)(x + 2) = (x + 2)2

9. Example
Determine whether each trinomial is a perfect square. If so, factor. If not explain.
Method 2 Use the rule.
x2 – 14x + 49
a = 1, b = 7
(x)2 – 2(x)(7) + 72
(x – 7)2
Write the trinomial as (a – b)2.

11. Example: Recognizing and Factoring the Difference of Two Squares
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.

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

13. Example
Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
x4 – 25y6
x4 – 25y6
5y3 5y3 
x2 x2
(x2 + 5y3)(x2 – 5y3)

14. Example
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)

15. Example
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)

16. Example
Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.
16x2 – 4y5
16x2 – 4y5
4x  4x
4y5 is not a perfect square.

17. Lesson Quiz: Part I
Determine whether each trinomial is a perfect square. If so factor. If not, explain.
64x2 – 40x + 25
2. 121x2 – 44x + 4
3. 49x x + 100

18. Lesson Quiz: Part II
Determine whether the binomial is a difference of two squares. If so, factor. If not, explain.
5. 9x2 – y4
6. 30x2 – y2
7. x2 – y8