1. 2.2 Polynomial Function of Higher Degrees
Zero of the function
Intermediate Value Theorem

2. Polynomial functions “Continuous” meaning no holes or breaks
Polynomial graphs have no sharp turns

3. Would a absolute value function be a polynomial?

4. Would a absolute value function be a polynomial?
Not a polynomial function

5. f(x) = xn is a Power function
If n is even If n is odd
The larger n, the flatter it is by the origin.

6. Ending behavior as x → ∞ If n is even and an > 0, as x → - ∞

7. Ending behavior as x → ∞ If n is odd and an > 0, as x → - ∞

8. Zeros of a Polynomial function
Numbers in the domain that make f(x) = 0.
We can the zeros with the factoring,
the Quadratic formula or Synthesis Division.
Find the zeros of g(x) = x 4 – 2x3 – 63x2
What is another name for a zero of a function?

9. Zeros of a Polynomial function
Numbers in the domain that make f(x) = 0.
We can the zeros with the factoring,
the Quadratic formula or Synthesis Division.
Find the zeros of g(x) = x 4 – 2x3 – 63x2
What is another name for a zero of a function?
Roots (lets call r) : so (x – r)(x – r)(x – r)… depends on degrees

10. g(x) = x 4 – 2x3 – 63x2

11. g(x) = x 4 – 2x3 – 63x2 It factors into x2(x + 7)(x – 9)
Set each to zero x2 = 0 x = 0 (d.r.)
x + 7 = 0 x = - 7
x – 9 = 0 x = 9
How would this help you graph?

12. What is the ending behavior?

13. What is the ending behavior?

14. What is the ending behavior?
What is happening between the zeros?
put numbers in g(x) between the zeros
to find out it the function is
positive or negative

15. Testing for positive or negative
Between – 7 and 0 lets use – 1
Between 0 and 9 lets use 1
g( -1) = (-1)2( )(- 1 – 9) = (1)(6)(-10)
negative
g(1) = (1)2(1 + 7)(1 – 9) = (1)(8)(-8)
In both cases the graph is below the x axis.

16. What is the ending behavior?
It gives the idea of what the graph should look like.

17. The Intermediate Value Theorem
Let a and b be real numbers such that a < b. If “f” is a polynomial function such that f(a)≠ f(b), then in the interval [a,b] ,“f” takes on every value between f(a) and f(b).
What does this do?

18. The Intermediate Value Theorem
Let a and b be real numbers such that a < b. If “f” is a polynomial function such that f(a)≠ f(b), then in the interval [a,b] ,“f” takes on every value between f(a) and f(b).
What does this do?
When f(x) goes from positive to negative or negative to positive, there exist one zero.

19. Find the intervals ( of 1 unit) where there is a zero of the function
f(x) = x4 + x3 – 7
f(- 3) = (- 3)4 + (-3)3 – 7 = 81 – 27 – 7
f(-3) = 47
f(-2) = 1
f(- 1) = -7 zero between – 2 and - 1
f(0) = -7
f(1) = - 5
f(2) = 17 zero between 1 and 2

20. Proof of the existence of √2
f(x) = x2 – 2
f(1) = (1)2 – 2 = - 1
f(2) = (2)2 – 2 = 2 There is a zero between
[ 1, 2]
Thus 0 = x2 – 2; 2 = x2 x = √2

21. Homework
Page
# 1 – 8, 15, 31, 38, 51, 63, 78, 89, 100, 108

22. Homework
Page 130 – 133
# 10, 20, 34, 42,
55, 71, 87, 93,
104, 112