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# Synthetic Division

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Synthetic Division

Synthetic Division

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• 1. Extra Section Synthetic Division Fo r us e w it h li nea r fact ors
• 2. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2
• 3. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 x − 3 3x + 2x − x + 3 3 2
• 4. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 2 3x x − 3 3x + 2x − x + 3 3 2
• 5. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 2 3x x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2
• 6. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 2 3x x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2 11x − x 2
• 7. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 3x +11x 2 x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2 11x − x 2
• 8. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 3x +11x 2 x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2 11x − x 2 −(11x − 33x) 2
• 9. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 3x +11x 2 x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2 11x − x 2 −(11x − 33x) 2 32x + 3
• 10. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 3x +11x +32 2 x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2 11x − x 2 −(11x − 33x) 2 32x + 3
• 11. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 3x +11x +32 2 x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2 11x − x 2 −(11x − 33x) 2 32x + 3 −(32x − 96)
• 12. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 3x +11x +32 2 x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2 11x − x 2 −(11x − 33x) 2 32x + 3 −(32x − 96) 99
• 13. Warm-up Divide. (3x + 2x − x + 3) ÷ (x − 3) 3 2 3x +11x +32 2 x − 3 3x + 2x − x + 3 3 2 −(3x − 9x ) 3 2 3x + 11x + 32, R : 99 2 11x − x 2 −(11x − 33x) 2 32x + 3 −(32x − 96) 99
• 14. Rational Roots Theorem
• 15. Rational Roots Theorem Let p be all factors of the leading coefficient and q be all factors of the constant in any polynomial. Then p/q gives all possible roots of the polynomial.
• 16. Synthetic Division
• 17. Synthetic Division Another way to divide polynomials, without the use of variables
• 18. Synthetic Division Another way to divide polynomials, without the use of variables Only works if you’re dividing by a linear factor
• 19. Synthetic Division Another way to divide polynomials, without the use of variables Only works if you’re dividing by a linear factor Allows for us to test whether a possible root is an actual zero
• 20. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2
• 21. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5
• 22. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5
• 23. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4
• 24. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4
• 25. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4
• 26. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 4
• 27. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 4 4
• 28. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 4 4 1
• 29. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 4 4 1
• 30. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 4 4 1 1
• 31. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 1 4 4 1 1
• 32. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 1 4 4 1 1 2
• 33. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 1 2 4 4 1 1 2
• 34. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 1 2 4 4 1 1 2 2
• 35. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 1 2 2 4 4 1 1 2 2
• 36. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 1 2 2 4 4 1 1 2 2 7
• 37. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 1 2 2 4 4 1 1 2 2 7
• 38. Example 1 Determine whether 1 is a root of 4x − 3x + x + 5 6 4 2 1 4 0 −3 0 1 0 5 4 4 1 1 2 2 4 4 1 1 2 2 7 4x + 4x + x + x + 2x + 2, R : 7 5 4 3 2
• 39. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2
• 40. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4
• 41. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5
• 42. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5 4
• 43. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5 5 4
• 44. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5 5 4 -2
• 45. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5 5 5 −2 4 -2
• 46. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5 5 5 −2 27 4 -2 − 2
• 47. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5 5 135 5 −2 − 8 27 4 -2 − 2
• 48. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5 5 135 5 −2 − 8 27 95 4 -2 − 2 − 8
• 49. Example 2 Use synthetic division to find the quotient and remainder. (4x − 7x − 11x + 5) ÷ (4x − 5) 3 2 5 4x − 5 → x − 4 5 4 4 −7 −11 5 5 135 5 −2 − 8 27 95 4 -2 − 2 − 8 27 95 4x − 2x − 2 2 ,R:− 8
• 50. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2
• 51. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3
• 52. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6
• 53. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6 6
• 54. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6 4 6
• 55. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6 4 6 -12
• 56. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6 4 -8 6 -12
• 57. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6 4 -8 6 -12 9
• 58. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6 4 -8 6 6 -12 9
• 59. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6 4 -8 6 6 -12 9 0
• 60. Example 3 Use synthetic division to find the quotient and remainder. (6x − 16x + 17x − 6) ÷ (3x − 2) 3 2 3x − 2 → x − 2 3 2 3 6 −16 17 −6 4 -8 6 6 -12 9 0 6x − 12x + 9, R : 0 2
• 62. Factoring a Quadratic Multiply a and c
• 63. Factoring a Quadratic Multiply a and c Factor ac into two factors that add up to b
• 64. Factoring a Quadratic Multiply a and c Factor ac into two factors that add up to b Replace b with these two values
• 65. Factoring a Quadratic Multiply a and c Factor ac into two factors that add up to b Replace b with these two values Group first 2 and last 2 terms
• 66. Factoring a Quadratic Multiply a and c Factor ac into two factors that add up to b Replace b with these two values Group first 2 and last 2 terms Factor out the GCF of each
• 67. Factoring a Quadratic Multiply a and c Factor ac into two factors that add up to b Replace b with these two values Group first 2 and last 2 terms Factor out the GCF of each Factors: (Stuff inside)(Stuff outside)
• 68. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2
• 69. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6
• 70. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12
• 71. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 = 4(−3)
• 72. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 = 4(−3) 2x + 4x − 3x − 6 2
• 73. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 = 4(−3) 2x + 4x − 3x − 6 2 (2x + 4x) + (−3x − 6) 2
• 74. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 = 4(−3) 2x + 4x − 3x − 6 2 (2x + 4x) + (−3x − 6) 2 2x(x + 2) − 3(x + 2)
• 75. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 = 4(−3) 2x + 4x − 3x − 6 2 (2x + 4x) + (−3x − 6) 2 2x(x + 2) − 3(x + 2) (x + 2)(2x − 3)
• 76. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 4i12 = 48 = 4(−3) 2x + 4x − 3x − 6 2 (2x + 4x) + (−3x − 6) 2 2x(x + 2) − 3(x + 2) (x + 2)(2x − 3)
• 77. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 4i12 = 48 = 4(−3) = (−16)(−3) 2x + 4x − 3x − 6 2 (2x + 4x) + (−3x − 6) 2 2x(x + 2) − 3(x + 2) (x + 2)(2x − 3)
• 78. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 4i12 = 48 = 4(−3) = (−16)(−3) 2x + 4x − 3x − 6 2 4x − 16x − 3x + 12 2 (2x + 4x) + (−3x − 6) 2 2x(x + 2) − 3(x + 2) (x + 2)(2x − 3)
• 79. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 4i12 = 48 = 4(−3) = (−16)(−3) 2x + 4x − 3x − 6 2 4x − 16x − 3x + 12 2 (2x + 4x) + (−3x − 6) 2 (4x − 16x) + (−3x + 12) 2 2x(x + 2) − 3(x + 2) (x + 2)(2x − 3)
• 80. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 4i12 = 48 = 4(−3) = (−16)(−3) 2x + 4x − 3x − 6 2 4x − 16x − 3x + 12 2 (2x + 4x) + (−3x − 6) 2 (4x − 16x) + (−3x + 12) 2 2x(x + 2) − 3(x + 2) 4x(x − 4) − 3(x − 4) (x + 2)(2x − 3)
• 81. Example 4 Factor. a. 2x + x − 6 2 b. 4x − 19x + 12 2 2i−6 = −12 4i12 = 48 = 4(−3) = (−16)(−3) 2x + 4x − 3x − 6 2 4x − 16x − 3x + 12 2 (2x + 4x) + (−3x − 6) 2 (4x − 16x) + (−3x + 12) 2 2x(x + 2) − 3(x + 2) 4x(x − 4) − 3(x − 4) (x + 2)(2x − 3) (x − 4)(4x − 3)
• 82. Homework
• 83. Homework Worksheet!