2. Essential Questions
β’ How do you evaluate functions using
synthetic division?
β’ How do you determine whether a binomial is a
factor of a polynomial by using synthetic
division?
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
8. Remainder Theorem
If a polynomial P(x) is divided by x β r, the
remainder is a constant P(r)
P(x ) = Q(x )i(x β r )+P(r )
where Q(x) is a polynomial with degree one less
than P(x) and P(r) is the evaluated polynomial
11. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
12. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
13. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
14. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
15. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
16. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
72
17. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
72
67
18. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
72
67
402
19. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
72
67
402
410
20. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
72
67
402
410
2460
21. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
72
67
402
410
2460
2453
22. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
72
67
402
410
2460
2453
23. Example 1
f (x ) = 2x 4
β 5x 2
+ 8x β 7
Use synthetic substitution to ο¬nd f(6) if
2 0 β5 8 β76
2
12
12
72
67
402
410
2460
2453
f (6) = 2453
24. Example 2
S(x ) = 0.02x 4
β 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
β 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 β1993
26. Example 2
S(x ) = 0.02x 4
β 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 β1993
x = 37
41. Example 2
S(x ) = 0.02x 4
β 0.52x 3
+ 4.03x 2
+ 0.09x + 77.54
0.02 β0.52 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
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
43. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
44. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
45. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
46. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
7
47. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
7
21
48. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
7
21
6
49. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
7
21
6
18
50. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
7
21
6
18
0
51. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
7
21
6
18
0
52. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
7
21
6
18
0
(x β 3)(x 2
+ 7x + 6)
53. Example 3
x 3
+ 4x 2
β15x β18
Determine whether x β 3 is a factor of the
polynomial below. Then ο¬nd the remaining
factors of the polynomial.
1 4 β15 β183
1
3
7
21
6
18
0
(x β 3)(x 2
+ 7x + 6)
(x β 3)(x + 6)(x +1)