1. 8.5 Factoring Special Products
Algebra 1
8.5 Factoring Special Products

2. Learning Targets
Language Goal: Students will be able to describe perfect squares.
Math Goal: Students will be able to factor perfect squares and students will be able to factor the difference of two squares.
Essential Question: Why are perfect squares helpful when factoring trinomials?

3. Warm-up

4. Perfect Square A trinomial is a perfect square if:
- the first and last terms are perfect squares
- the middle term is two times one factor from the first term and one factor from the last term.
9x² x
3x 3x 2(3x • 2) 2 • 2

5. 𝑥 2 +6𝑥+9= 𝑥+3 𝑥+3 = (𝑥+3) 2
𝑥 2 −2𝑥+1= 𝑥−1 𝑥−1 = (𝑥−1) 2
𝑎 2 +2𝑎𝑏+ 𝑏 2 = 𝑎+𝑏 𝑎+𝑏 = (𝑎+𝑏) 2
𝑎 2 −2𝑎𝑏+ 𝑏 2 = 𝑎−𝑏 𝑎−𝑏 = (𝑎−𝑏) 2

6. Example Type 1: Recognizing and Factoring Perfect-Square Trinomials
Determine whether each trinomial is a perfect square. If so, factor. If not, explain. A. x² + 12x + 36 B. 4x² – 12x + 9

7. Example Type 1: Recognizing and Factoring Perfect-Square Trinomials
Determine whether each trinomial is a perfect square. If so, factor. If not, explain. C. x² + 9x + 16 D. x² + 4x + 4

8. Example Type 1: Recognizing and Factoring Perfect-Square Trinomials
Determine whether each trinomial is a perfect square. If so, factor. If not, explain. E. x² – 14x + 49 F. 9x² – 6x + 4

9. Example Type 1: Recognizing and Factoring Perfect-Square Trinomials
Determine whether each trinomial is a perfect square. If so, factor. If not, explain. G. 9x² – 15x + 64 H. 81x² + 90x + 25 I. 36x² – 10x + 14

10. Example 2: Word Applications
A. The park in the center of the Place des Vosges in Paris, France, is in the shape of a square. The area of the park is (25x² + 70x + 49) ft². The side length of the park is in the form cx + d, where c and d are whole numbers. Find an expression in terms of x for the perimeter of the park. Find the perimeter when x = 8 feet.

11. Example 2: Word Applications
B. A company produces square sheets of aluminum each of which has an area of (9x² + 6x + 1) m². The side length of each sheet is in the form cx + d, where c and d are whole numbers. Find an expression in terms of x for the perimeter of a sheet. Find the perimeter when x = 3 m.

12. Example 2: Word Applications
C. A rectangular piece of cloth must be cut to make a tablecloth. The area needed is (16x² - 24x + 9) ft². The dimensions of the cloth are in the form cx + d, where c and d are whole numbers. Find an expression in terms of x for the perimeter of the cloth. Find the perimeter when x = 11 in.

13. Difference of Two Squares
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

14. Difference of Two Squares
𝑎 2 − 𝑏 2 = 𝑎+𝑏 𝑎−𝑏
𝑥 2 −9= 𝑥+3 𝑥−3

15. Example Type 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.
A. x² – B. 9p – 16q²

16. Example Type 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.
C. x – 7y² D. 1 – 4x²

17. Example Type 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.
E. p – 49q F. 16x² – 4y

18. Example Type 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.
G. 3p² – 9q H. 100x² – 4y² I. x – 25y

19. Closure
(𝑥+1) 2
𝑥 2 +2𝑥+1
(𝑥−1) 2
𝑥 2 −2𝑥+1
𝑥+1 (𝑥−1)
𝑥 2 −1

20. Lesson Quiz