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Objective The student will be able to:

Objective The student will be able to:. factor using difference of squares factor perfect square trinomials. Difference of Squares Factoring Questions to ask Type Number of Terms. 1. GCF 2 or more 2. Difference of Squares 2. Determine the pattern. = 1 2

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Objective The student will be able to:

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  1. ObjectiveThe student will be able to: • factor using difference of squares • factor perfect square trinomials

  2. Difference of SquaresFactoring Questions to ask 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. Difference of Squares a2 - b2 = (a - b)(a + b)or a2 - b2 = (a + b)(a - b) The order does not matter!!

  5. 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!!!

  6. No 1. Factor x2 - 25 Yes Yes x2 – 25 Yes 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 - ( )( ) x + 5 x 5

  7. No 2. Factor 16x2 - 9 Yes 16x2 – 9 Yes 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 - (4x )(4x ) + 3 3

  8. 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 +

  9. Factor x2 – y2 (x – y)(x + y) OR (x + y)(x – y) Multiplication is communitive  order doesn’t matter

  10. 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 +

  11. Factor 18c2 + 8d2 GCF = 2 2(9c2 + 4d2) • Two terms? • 1st term a perfect square? • 2nd term a perfect square? • Subtraction? YES YES YES NO!!! You cannot factor using difference of squares because there is no subtraction! Final Answer: 2(9c2 + 4d2)

  12. Factor -64 + 4m2 Rewrite the problem as 4m2 – 64 so the subtraction is in the middle! 4m2 – 64 GCF? Yes => 4 Is the leftover the difference of perfect squares?? YES 4(m -4 )(m + 4) Factor =>

  13. Perfect Square TrinomialsFactoring Questions TypeNumber of Terms 1. GCF 2 or more 2. Diff. Of Squares 2 3. Trinomials 3

  14. Perfect Square Trinomials (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2

  15. 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

  16. 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!

  17. 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

  18. Factor m2 – 12m + 36 NO!! Any GCF? a and c perfect squares? YES!! Factors of first term = m * m Factors of last term = 6 * 6 (m - 6)(m - 6 )

  19. 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

  20. 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

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

  22. 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!

  23. Conditions for Difference of Squares • Must be a binomial with subtraction. • First term must be a perfect square. (x)(x) = x2 • Second term must be a perfect square (6)(6) = 36

  24. Recognizing the Difference of Squares Must be a binomial with subtraction. First term must be a perfect square (p)(p) = p2 Second term must be a perfect square (10)(10) = 100

  25. Recognizing the Difference of Squares Must be a binomial with subtraction. First term must be a perfect square (3m)(3m) = 9m2 Second term must be a perfect square (7)(7) = 49

  26. Check for GCF. Sometimes it is necessary to remove the GCF before it can be factored more completely.

  27. Removing a GCF of -1. In some cases removing a GCF of negative one will result in the difference of squares.

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