Synthetic Division

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!