Synthetic Division
Upcoming SlideShare
Loading in...5
×

Like this? Share it with your network

Share

Synthetic Division

  • 2,680 views
Uploaded on

Synthetic Division

Synthetic Division

More in: Education , Technology
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
2,680
On Slideshare
2,587
From Embeds
93
Number of Embeds
6

Actions

Shares
Downloads
114
Comments
0
Likes
0

Embeds 93

http://mrlambmath.wikispaces.com 74
http://www.slideshare.net 10
http://blackboard.cpsb.org 5
https://mrlambmath.wikispaces.com 2
http://bb.samhouston.cpsb.org 1
http://www.protopage.com 1

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 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
  • 61. Factoring a Quadratic
  • 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!