Objective The student will be able to:

1. ObjectiveThe student will be able to: factor using difference of squares.

2. Factoring ChartThis chart will help you to determine which method of factoring to use.TypeNumber of Terms 1. GCF 2 or more 2. Difference of Squares 2

3. Determine the pattern = 12 = 22 = 32 = 42 = 52 = 62 These are perfect squares! You should be able to list the first 15 perfect squares in 30 seconds… 1 4 9 16 25 36 … Perfect squares1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

4. First terms: Outer terms: Inner terms: Last terms: Combine like terms. x2 – 4 Review: Multiply (x – 2)(x + 2) Notice the middle terms eliminate each other! x2 +2x -2x x2 -2x -4 +2x -4 This is called the difference of squares.

5. Difference of Squares a2 - b2 = (a - b)(a + b)or a2 - b2 = (a + b)(a - b) The order does not matter!!

6. 4 Steps for factoringDifference of Squares 1. Are there only 2 terms? 2. Is the first term a perfect square? 3. Is the last term a perfect square? 4. Is there subtraction (difference) in the problem? If all of these are true, you can factor using this method!!!

7. No 1. Factor x2 - 25 x2 – 25 Yes When factoring, use your factoring table. Do you have a GCF? Are the Difference of Squares steps true? Two terms? 1st term a perfect square? 2nd term a perfect square? Subtraction? Write your answer! Yes Yes - Yes ( )( ) x + 5 x 5

8. No 2. Factor 16x2 - 9 16x2 – 9 Yes When factoring, use your factoring table. Do you have a GCF? Are the Difference of Squares steps true? Two terms? 1st term a perfect square? 2nd term a perfect square? Subtraction? Write your answer! Yes Yes - Yes (4x )(4x ) + 3 3

9. No 3. Factor 81a2 – 49b2 81a2 – 49b2 Yes When factoring, use your factoring table. Do you have a GCF? Are the Difference of Squares steps true? Two terms? 1st term a perfect square? 2nd term a perfect square? Subtraction? Write your answer! Yes Yes - Yes (9a )(9a ) 7b 7b +

10. Factor x2 – y2 • (x + y)(x + y) • (x – y)(x + y) • (x + y)(x – y) • (x – y)(x – y) Remember, the order doesn’t matter!

11. Yes! GCF = 3 4. Factor 75x2 – 12 Yes 3(25x2 – 4) When factoring, use your factoring table. Do you have a GCF? 3(25x2 – 4) Are the Difference of Squares steps true? Two terms? 1st term a perfect square? 2nd term a perfect square? Subtraction? Write your answer! Yes Yes Yes - 3(5x )(5x ) 2 2 +

12. Factor 18c2 + 8d2 • prime • 2(9c2 + 4d2) • 2(3c – 2d)(3c + 2d) • 2(3c + 2d)(3c + 2d) You cannot factor using difference of squares because there is no subtraction!

13. Factor -64 + 4m2 • prime • (2m – 8)(2m + 8) • 4(-16 + m2) • 4(m – 4)(m + 4) Rewrite the problem as 4m2 – 64 so the subtraction is in the middle!

14. ObjectiveThe student will be able to: factor perfect square trinomials.

15. Factoring ChartThis chart will help you to determine which method of factoring to use.TypeNumber of Terms 1. GCF 2 or more 2. Diff. Of Squares 2 3. Trinomials 3

16. Review: Multiply (y + 2)2(y + 2)(y + 2) Do you remember these? (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 y2 First terms: Outer terms: Inner terms: Last terms: Combine like terms. y2 + 4y + 4 Using the formula, (y + 2)2 = (y)2 + 2(y)(2) + (2)2 (y + 2)2 = y2 + 4y + 4 Which one is quicker? +2y +2y +4

17. 1) Factor x2 + 6x + 9 Perfect Square Trinomials (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 Does this fit the form of our perfect square trinomial? • Is the first term a perfect square? Yes, a = x 2) Is the last term a perfect square? Yes, b = 3 • Is the middle term twice the product of the a and b? Yes, 2ab = 2(x)(3) = 6x Since all three are true, write your answer! (x + 3)2 You can still factor the other way but this is quicker!

18. 2) Factor y2 – 16y + 64 Perfect Square Trinomials (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 Does this fit the form of our perfect square trinomial? • Is the first term a perfect square? Yes, a = y 2) Is the last term a perfect square? Yes, b = 8 • Is the middle term twice the product of the a and b? Yes, 2ab = 2(y)(8) = 16y Since all three are true, write your answer! (y – 8)2

19. Factor m2 – 12m + 36 • (m – 6)(m + 6) • (m – 6)2 • (m + 6)2 • (m – 18)2

20. 3) Factor 4p2 + 4p + 1 Perfect Square Trinomials (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 Does this fit the form of our perfect square trinomial? • Is the first term a perfect square? Yes, a = 2p 2) Is the last term a perfect square? Yes, b = 1 • Is the middle term twice the product of the a and b? Yes, 2ab = 2(2p)(1) = 4p Since all three are true, write your answer! (2p + 1)2

21. Perfect Square Trinomials (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 4) Factor 25x2 – 110xy + 121y2 Since all three are true, write your answer! (5x – 11y)2 Does this fit the form of our perfect square trinomial? Is the first term a perfect square? Yes, a = 5x Is the last term a perfect square? Yes, b = 11y Is the middle term twice the product of the a and b? Yes, 2ab = 2(5x)(11y) = 110xy

22. Factor 9k2 + 12k + 4 • (3k + 2)2 • (3k – 2)2 • (3k + 2)(3k – 2) • I’ve got no clue…I’m lost!

23. Factor 2r2 + 12r + 18 • prime • 2(r2 + 6r + 9) • 2(r – 3)2 • 2(r + 3)2 • 2(r – 3)(r + 3) Don’t forget to factor the GCF first!