This document discusses synthetic division and the remainder and factor theorems. It provides examples of using synthetic division to evaluate functions, determine the number of terms in a sequence, and factor polynomials. The key steps of synthetic division are shown, along with checking the remainder and determining common factors. Three examples are worked through to demonstrate these concepts.

1. Section 4-8
The Remainder and Factor
Theorem

2. Essential Questions
• How do you evaluate functions using
synthetic division?
• How do you determine whether a binomial is a
factor of a polynomial by using synthetic
division?

3. Vocabulary
1. Synthetic Substitution:
2. Depressed Polynomial:

4. Vocabulary
1. Synthetic Substitution: Using synthetic division
to determine the solution to evaluating a
function
2. Depressed Polynomial:

5. Vocabulary
1. Synthetic Substitution: Using synthetic division
to determine the solution to evaluating a
function
2. Depressed Polynomial: A polynomial that is
generally unhappy

6. Vocabulary
1. Synthetic Substitution: Using synthetic division
to determine the solution to evaluating a
function
2. Depressed Polynomial: A polynomial that is one
degree less than the original polynomial

7. Remainder Theorem

8. Remainder Theorem
If a polynomial P(x) is divided by x − r, the
remainder is a constant P(r)
P(x ) = Q(x )i(x − r )+P(r )
where Q(x) is a polynomial with degree one less
than P(x) and P(r) is the evaluated polynomial

9. Factor Theorem

10. Factor Theorem
The binomial x − r is a factor of the polynomial
P(x) IFF P(r) =0

11. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if

12. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76

13. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2

14. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12

15. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12

16. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12
72

17. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12
72
67

18. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12
72
67
402

19. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12
72
67
402
410

20. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12
72
67
402
410
2460

21. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12
72
67
402
410
2460
2453

22. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12
72
67
402
410
2460
2453

23. Example 1
f (x ) = 2x 4
− 5x 2
+ 8x − 7
Use synthetic substitution to find f(6) if
2 0 −5 8 −76
2
12
12
72
67
402
410
2460
2453
f (6) = 2453

24. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
The number of college students from the United
States who study abroad can be modeled by
the function below, where x is the number of
years since 1993 and S(x) is the number of
students in thousands. Find the number of
college students from the US that will study
abroad in 2030.

25. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
The number of college students from the United
States who study abroad can be modeled by
the function below, where x is the number of
years since 1993 and S(x) is the number of
students in thousands. Find the number of
college students from the US that will study
abroad in 2030.
x = 2030 −1993

26. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
The number of college students from the United
States who study abroad can be modeled by
the function below, where x is the number of
years since 1993 and S(x) is the number of
students in thousands. Find the number of
college students from the US that will study
abroad in 2030.
x = 2030 −1993
x = 37

27. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54

28. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437

29. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02

30. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74

31. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22

32. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14

33. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17

34. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17
450.29

35. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17
450.29
450.38

36. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17
450.29
450.38
16664.06

37. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17
450.29
450.38
16664.06
16741.6

38. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17
450.29
450.38
16664.06
16741.6

39. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17
450.29
450.38
16664.06
16741.6
16741.6 i1000

40. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17
450.29
450.38
16664.06
16741.6
16741.6 i1000
16741600

41. Example 2
S(x ) = 0.02x 4
− 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 −0.52 4.03 0.09 77.5437
0.02
0.74
0.22
8.14
12.17
450.29
450.38
16664.06
16741.6
16741.6 i1000
16741600
There will be 16,741,600 students studying
abroad in 2030.

42. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.

43. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183

44. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1

45. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3

46. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3
7

47. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3
7
21

48. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3
7
21
6

49. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3
7
21
6
18

50. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3
7
21
6
18
0

51. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3
7
21
6
18
0

52. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3
7
21
6
18
0
(x − 3)(x 2
+ 7x + 6)

53. Example 3
x 3
+ 4x 2
−15x −18
Determine whether x − 3 is a factor of the
polynomial below. Then find the remaining
factors of the polynomial.
1 4 −15 −183
1
3
7
21
6
18
0
(x − 3)(x 2
+ 7x + 6)
(x − 3)(x + 6)(x +1)