1. Chapter 3
Polynomial & Rational
Functions
Romans 8:6 For to set the mind on the flesh is
death, but to set the mind on the Spirit is life
and peace.

2. 3.1 Polynomial Functions and Their Graphs

3. 3.1 Polynomial Functions and Their Graphs
Standard Form of a polynomial

4. 3.1 Polynomial Functions and Their Graphs
Standard Form of a polynomial
n n−1 n−2
P(x) = an x + an−1 x + an−2 x + ... + a1 x + a0
an ≠ 0

5. Review the graphs of polynomial functions
of increasing degree

6. Review the graphs of polynomial functions
of increasing degree
Degree # Extrema
0 NA
1 NA
2 1
3 2
4 3
5 4
n n-1

7. Review the graphs of polynomial functions
of increasing degree
Degree # Extrema
0 NA
1 NA
2 1
3 2
4 3
5 4
n n-1

8. Review the graphs of polynomial functions
of increasing degree
Degree # Extrema
0 NA
1 NA
2 1
3 2
4 3
5 4
n n-1

9. Review the graphs of polynomial functions
of increasing degree
Degree # Extrema
0 NA
1 NA
2 1
3 2
4 3
5 4
n n-1

10. Review the graphs of polynomial functions
of increasing degree
Degree # Extrema
0 NA
1 NA
2 1
3 2
4 3
5 4
n n-1

11. Review the graphs of polynomial functions
of increasing degree
Degree # Extrema
0 NA
1 NA
2 1
3 2
4 3
5 4
n n-1

12. Review the graphs of polynomial functions
of increasing degree
Degree # Extrema
0 NA
1 NA
2 1
3 2
4 3
5 4
n n-1

13. Review the graphs of polynomial functions
of increasing degree
Degree # Extrema
0 NA
1 NA
2 1
3 2
4 3
5 4
n n-1

14. End Behavior

15. End Behavior
How does the graph behave as x gets
very large or very small ...

16. End Behavior
How does the graph behave as x gets
very large or very small ...
x→∞
x approaches positive infinity (right end)

17. End Behavior
How does the graph behave as x gets
very large or very small ...
x→∞
x approaches positive infinity (right end)
x → −∞
x approaches negative infinity (left end)

18. The end behavior of a polynomial function
is determined by the term that contains
the highest power of x (the variable)

19. The end behavior of a polynomial function
is determined by the term that contains
the highest power of x (the variable)
The other terms become insignificant as
x→∞

20. Use a grapher to show that
3 2
f (x) = 8x + 7x + 3x + 7
and f (x) = 8x have the same end behavior
3

21. Use a grapher to show that
3 2
f (x) = 8x + 7x + 3x + 7
and f (x) = 8x have the same end behavior
3
as x → ∞, y → ∞
as x → −∞, y → −∞

22. Real Zeros of Polynomials
If P is a polynomial and c is a real
number and c is a zero of P
Then

23. Real Zeros of Polynomials
If P is a polynomial and c is a real
number and c is a zero of P
Then
1) x = c is a solution of P(x) = 0

24. Real Zeros of Polynomials
If P is a polynomial and c is a real
number and c is a zero of P
Then
1) x = c is a solution of P(x) = 0
2) (x − c) is a factor of P(x)

25. Real Zeros of Polynomials
If P is a polynomial and c is a real
number and c is a zero of P
Then
1) x = c is a solution of P(x) = 0
2) (x − c) is a factor of P(x)
3) x = c is an x-intercept of P(x)

26. Graph: y = x(x − 4)(x + 2)

27. Graph: y = x(x − 4)(x + 2)
3 x-intercepts (3 zeros)

28. Graph: y = x(x − 4)(x + 2)
3 x-intercepts (3 zeros)
3 solutions to 0 = x(x − 4)(x + 2)

29. Graph: y = x(x − 4)(x + 2)
3 x-intercepts (3 zeros)
3 solutions to 0 = x(x − 4)(x + 2)
This is a cubic function

30. Intermediate Value Theorem
If P is a polynomial and P(a) and P(b)
have opposite signs, then there is at least
one value c between a and b such that
P(c)=0.

31. Intermediate Value Theorem
If P is a polynomial and P(a) and P(b)
have opposite signs, then there is at least
one value c between a and b such that
P(c)=0.

32. Multiplicity of Roots

33. Multiplicity of Roots
Consider: y = (x − 3)(x − 3)(x + 1)

34. Multiplicity of Roots
Consider: y = (x − 3)(x − 3)(x + 1)
Zeros are:

35. Multiplicity of Roots
Consider: y = (x − 3)(x − 3)(x + 1)
Zeros are: x = 3, x = 3, x = −1

36. Multiplicity of Roots
Consider: y = (x − 3)(x − 3)(x + 1)
Zeros are: x = 3, x = 3, x = −1
Multiplicity of 2

37. Multiplicity of Roots
Consider: y = (x − 3)(x − 3)(x + 1)
Zeros are: x = 3, x = 3, x = −1
Multiplicity of 2

38. Multiplicity of Roots
Consider: y = (x − 3)(x − 3)(x + 1)
Zeros are: x = 3, x = 3, x = −1
Multiplicity of 2
Graph the function to “see” the multiplicity

39. Multiplicity of Roots

40. Multiplicity of Roots
When a graph “bounces” ... even multiplicity
(2, 4, 6, ... )
When a graph flattens out and goes
through the axis ... odd multiplicity
(3, 5, 7, ...)

41. Multiplicity of Roots
When a graph “bounces” ... even multiplicity
(2, 4, 6, ... )
When a graph flattens out and goes
through the axis ... odd multiplicity
(3, 5, 7, ...)
Graph: y = (x + 3)(x + 3)(x + 3)(x − 4)

42. HW #1
“If we are to go only halfway or reduce our
sights in the face of difficulty ... it would be
better to not go at all.” John F. Kennedy