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Perfect Square Trinomials Factoring Techniques

Learn how to factor perfect square trinomials efficiently with step-by-step examples and tips for using the difference of squares method. Practice solving equations using the zero product property.

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Perfect Square Trinomials Factoring Techniques

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  1. Bell Ringer 2/20/15 Completely Factor & Check your answer. • Factor: 2x2- 14x + 12 • Factor: y2 + 4y + 4 • Factor: 75x2– 12

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

  3. Factoring ChartThis chart will help you to determine which method of factoring to use.TypeNumber of Terms 1. GCF 2 or more 2. Grouping 4 3. Trinomials 3

  4. Review: Multiply (y + 2)2(y + 2)(y + 2) Check this out…whaaat!! (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

  5. 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!

  6. 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

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

  8. 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

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

  10. Factoring ChartThis chart will help you to determine which method of factoring to use.TypeNumber of Terms 1. GCF 2 or more 2. Grouping 4 3. Trinomials 3 4. Difference of Squares 2

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

  12. 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.

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

  14. 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!!!

  15. 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

  16. 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

  17. 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 +

  18. 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!

  19. 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!

  20. ObjectiveThe student will be able to: use the zero product property to solve equations

  21. Zero Product Property If a • b = 0 then a=0, b=0, or both a and b equal 0.

  22. 4 steps for solving a quadratic equation Set = 0 Factor Split/Solve Check • Set the equation equal to 0. • Factor the equation. • Set each part equal to 0 and solve. • Check your answer on the calculator if available.

  23. 1. Solve (x + 3)(x - 5) = 0 Using the Zero Product Property, you know that either x + 3 = 0or x - 5 = 0 Solve each equation. x = -3 or x = 5 {-3, 5}

  24. 2. Solve (2a + 4)(a + 7) = 0 2a + 4 = 0 or a + 7 = 0 2a = -4 or a = -7 a = -2 or a = -7 {-2, -7}

  25. 3. Solve (3t + 5)(t - 3) = 0 3t + 5 = 0 or t - 3 = 0 3t = -5 or t = 3 t = -5/3 or t = 3 {-5/3, 3}

  26. 4. Solve x2 - 11x = 0 Set = 0 Factor Split/Solve Check GCF = x x(x - 11) = 0 x = 0 or x - 11 = 0 x = 0 or x = 11 {0, 11}

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