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Objective: The Learner will..,

9.6 Factoring Special Polynomials. Objective: The Learner will..,. Factor the difference of two perfect squares. Factor perfect-square trinomials. Factor trinomials into two binomials using factor/sum tables. NCSCOS. 1.01, 1.02, and 4.02. 9.6 Factoring Special Polynomials.

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Objective: The Learner will..,

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  1. 9.6 Factoring Special Polynomials Objective: The Learner will.., • Factor the difference of two perfect squares. Factor perfect-square trinomials. • Factor trinomials into two binomials using factor/sum tables NCSCOS • 1.01, 1.02, and 4.02

  2. 9.6 Factoring Special Polynomials Review (9-2 & 9-3): (a – b)(a + b) (a – b)2 (a + b)2 = (a – b)(a – b) = (a + b)(a + b) = a2–b2 = a2– 2ab + b2 = a2 + 2ab + b2 Simplify: a2–b2 a2– 2ab + b2 a2 + 2ab + b2 = (a – b)(a – b) = (a + b)(a + b) = (a – b)2 = (a – b)(a + b) = (a + b)2

  3. 9.6 Factoring Special Polynomials Factor each expression: Solution: Take the square root of the first and last terms; then write two binomials, with the two solutions; one with a plus and the other with a minus: x2 – 100 (x – 10)(x + 10) x2– 36 x2 – 25 x2 – 9 (x– 6)(x + 6) (x – 5)(x + 5) (x – 3)(x + 3) x2– 49 x2 – 4 x2 – 16 (x– 7)(x + 7) (x – 2)(x + 2) (x – 4)(x + 4)

  4. 9.6 Factoring Special Polynomials Factor each expression: x2– 1 25x2– y2 (5x– y)(5x + y) 9x2– 25 (x– 1)(x + 1) (3x– 5)(3x + 5) x2 – 64 36x2 – 64y2 16x2 – 36 (x – 8)(x + 8) (6x – 8y)(6x + 8y) (4x – 6)(4x + 6) x2 – 81 100x2 – 121 (x – 9)(x + 9) (10x – 11)(10x + 11) 4x2 – 16 (2x – 4)(2x + 4)

  5. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) x2 + 5x + 6 (middle term) (1)(6) = 6 1 + 6 = 7 x2 + 2x + 3x + 6 (2)(3) = 6 2 + 3 = 5 (x2 + 2x) + (3x + 6) x(x + 2) + 3(x + 2) = (x + 3)(x + 2)

  6. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) x2 + 14x + 49 (middle term) (1)(49) = 49 1 + 49 = 50 x2 + 7x + 7x + 49 (7)(7) = 49 7 + 7 = 14 (x2 + 7x) + (7x + 49) x(x + 7) + 7(x + 7) (x + 7)(x + 7) = (x + 7)2

  7. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (middle term) (last term) x2– 8x + 16 –1– 16 = –17 (–1)(–16) = 16 x2– 4x – 4x + 16 –2 – 8 = –10 (–2)(–8) = 16 (x2– 4x) + (–4x + 16) –4 – 4 = –8 (–4)(–4) = 16 x(x – 4) – 4(x – 4) (x – 4)(x – 4) = (x – 4)2

  8. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) x2 + 10x + 25 (middle term) (1)(25) = 25 1 + 25 = 26 x2 + 5x + 5x + 25 (5)(5) = 25 5 + 5 = 10 (x2 + 5x) + (5x + 25) x(x + 5) + 5(x + 5) (x + 5)(x + 5) = (x + 5)2

  9. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) (middle term) x2– 12x + 36 (–1)(–36) = 36 –1 – 36 = –37 x2– 6x – 6x + 36 –2 – 18 = –20 (–2)(–18) = 36 (x2– 6x) + (6x – 6) (–3)(–12) = 36 –3 – 12 = –15 x(x – 6) – 6(x – 6) (–4)(–9) = 36 –4 –9 = –13 (x – 6)(x – 6) (–6)(–6) = 36 –6 – 6 = –12 = (x – 6)2

  10. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) x2 + 8x + 16 (middle term) (1)(16) = 16 1 + 16 = 17 x2 + 4x + 4x + 16 (2)(8) = 16 2 + 8 = 10 (x2 + 4x) + (4x + 16) (4)(4) = 16 4 + 4 = 8 x(x + 4) + 4(x + 4) (x + 4)(x + 4) = (x + 4)2

  11. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (middle term) (last term) x2– 10x + 25 –1– 25 = –26 (–1)(–25) = 25 x2– 5x – 5x + 25 (–5)(–5) = 25 –5 – 5 = –10 (x2– 5x) + (–5x + 25) x(x – 5) – 5(x – 5) (x – 5)(x – 5) = (x – 5)2

  12. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) (middle term) x2 + 6x + 9 (1)(9) = 9 1 + 9 = 10 x2 + 3x + 3x + 9 (3)(3) = 9 3 + 3 = 6 (x2 + 3x) + (3x + 9) x(x + 3) + 3(x + 3) (x + 3)(x + 3) = (x + 3)2

  13. 9.6 Factoring Special Polynomials Factor each expression: x2– 18x + 81 x2 + 16x + 64 (x+ 8)2 (x– 9)2 x2 + 12x + 36 x2 + 18x + 81 (x+ 6)2 (x+ 9)2 x2 – 6x + 9 x2– 16x + 64 (x– 3)2 (x– 8)2

  14. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (middle term) (last term) x2– 18x + 81 –1– 49 = –50 (–1)(–81) = 81 x2– 9x – 9x + 81 –3 – 27 = –30 (–3)(–27) = 81 (x2– 9x) + (–9x + 81) (–9)(–9) = 81 –9 –9 = –18 x(x – 9) – 9(x – 9) (x – 9)(x – 9) = (x – 9)2

  15. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) x2 + 12x + 36 (middle term) (1)(36) = 36 1 + 36 = 37 x2 + 6x + 6x + 36 2 + 18 = 20 (2)(18) = 36 (x2 + 6x) + (6x + 36) (3)(12) = 36 3 + 12 = 15 x(x + 6) + 6(x + 6) (4)(9) = 36 4 + 9 = 13 (x + 6)(x + 6) (6)(6) = 36 6 + 6 = 12 = (x + 6)2

  16. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (middle term) (last term) x2– 6x + 9 –1– 9 = –10 (–1)(–9) = 9 x2– 3x – 3x + 9 –3 – 3 = –6 (–3)(–3) = 9 (x2– 3x) + (–3x + 9) x(x – 3) – 3(x – 3) (x – 3)(x – 3) = (x – 3)2

  17. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) x2 + 16x + 64 (middle term) (1)(64) = 64 1 + 64 = 65 x2 + 8x + 8x + 64 2 + 32 = 34 (2)(32) = 64 (x2 + 8x) + (8x + 64) (4)(16) = 64 4 + 16 = 20 x(x + 8) + 8(x + 8) (8)(8) = 64 8 + 8 = 16 (x + 8)(x + 8) = (x + 8)2

  18. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) x2 + 18x + 81 (middle term) (1)(81) = 81 1 + 49 = 50 x2 + 9x + 9x + 81 (3)(27) = 81 3 + 27 = 30 (x2 + 9x) + (9x + 81) 9 + 9 = 18 (9)(9) = 81 x(x + 9) + 9(x + 9) (x + 9)(x + 9) = (x + 9)2

  19. 9.6 Factoring Special Polynomials Factor using Factor/Sum Table (last term) x2– 16x + 64 (middle term) (–1)(– 64) = 64 –1 – 64 = –65 x2– 8x – 8x + 64 (–2)(–32) = 64 –2 – 32 = –34 (x2– 8x) + (–8x + 64) (–4)(–16) = 64 –4 – 16 = –20 x(x – 8) – 8(x – 8) (–8)(–8) = 64 –8 – 8 = –16 (x – 8)(x – 8) = (x – 8)2

  20. 9.6 Factoring Special Polynomials Factoring complex perfect square polynomials. 4x2 + 24xy + 36y2 1. Take the square root of the first and last terms. 2x 6y 2. Close with parentheses, list the correct sign, and raise to a power of 2. (2x + 6y)2 Check: Divide the middle term by two; is this the product of the numbers in parentheses? 24xy = 12xy 2 If so, you are finished checking; if not, try again. (2x)(6y) = 12xy

  21. 9.6 Factoring Special Polynomials Factor each expression: x2– 12xy + 36y2 100m2–140m + 49 (10m– 7)2 (x– 6y)2 64a4– 16a2b + b2 9s2 + 24s + 16 (8a2– b)2 (3s + 4)2 4x2 + 12x + 9 36a2– 24ab + 4b2 (2x+ 3)2 (6a2– 2b)2

  22. 9.6 Factoring Special Polynomials Factor each expression: a2x2 + 2abx + b2 49x2–42x + 9y2 (7x – 3y)2 (ax+ b)2 36d2 + 12d + 1 16x2 + 72x + 81y2 (6d + 1)2 (4x + 9y)2 9a2– 12a + 4 36a2– 60a + 25 (3a – 2)2 (6a2– 5)2

  23. 9.6 Factoring Special Polynomials 24x 16 Find the missing term, then find the PS binomial: x2 + 8x + ____ x2 + ____ + 144 x2 – 6x + ____ x2 + ____ + 49 x2 + 10x + ____x2 – ____ + 36 (x + 4)2 (x + 12)2 9 14x (x + 7)2 (x – 3)2 25 12x (x – 6)2 (x + 5)2

  24. 9.6 Factoring Special Polynomials 22x 4 Find the missing term: x2 + 4x + ____ x2 + ____ + 121 x2 – 16x + ____ x2 + ____ + 100 x2 + 18x + ____x2 –____+ 1 (x + 2)2 (x + 11)2 64 20x (x + 10)2 (x – 8)2 81 2x (x – 1)2 (x + 9)2

  25. 9.6 Factoring Special Polynomials Algebraic relationships of PS binomials: 2 b2 b2 a2 + 2ab + b2 = x2 + bx + = (x + )2 = (a + b)2 2 82 82 x2 + 8x + 16 = x2 + 8x + = (x + )2 = (x + 4)2 = (x + 4)2

  26. 9.6 Factoring Special Polynomials Algebraic relationships of PSBs: 2 -b b2 a2– 2ab + b2 = x2– bx + = (x – )2 2 = (a – b)2 2 -14 14 = x2– 14x + x2– 14x + 49 = (x – )2 2 2 = (x – 7)2 = (x – 7)2

  27. 9.6 Factoring Special Polynomials Algebraic relationships of PS binomials: 2 122 122 x2 + 12x + 36 = x2 + 12x + = (x + )2 = (x + 6)2 2 12 12 x2– 12x + 36 = x2– 12x + = (x – )2 2 2 = (x – 6)2

  28. 9.6 Factoring Special Polynomials Factor Completely: (x2 – 4)(x2 – 9) =(x – 2)(x + 2)(x – 3)(x + 3) (x4 – 16)(x2 – 36) = (x2 – 4)(x2 + 4)(x + 6)(x – 6) = (x– 2)(x + 2)(x2 + 4)(x + 6)(x – 6) (x4 – 16)(36 – x2) = (x2 – 4)(x2 + 4)(6 + x)(6 – x) = (x– 2)(x + 2)(x2 + 4)(6 + x)(6 – x)

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