1. Polynomial Functions and Models
Section 5.1
Polynomial Functions and Models 1

2. Polynomial Functions
Three of the families of functions studied thus far:
constant, linear, and quadratic, belong to a much
larger group of functions called polynomials.
We begin our formal study of general polynomials
with a definition and some examples.

3. f (x)  an xn + an1 xn1 + … + a2 x2 + a1 x + a0
Polynomial Functions
A polynomial function is a function of the form
f (x)  an xn + an1 xn1 + … + a2 x2 + a1 x + a0
where a0, a1, , an are real numbers and n  1 is
a natural number.
The domain of a polynomial function is ( , ).

4. f (x)  an xn + an1 xn1 + … + a2 x2 + a1 x + a0
Polynomial Functions
Suppose f is the polynomial function
f (x)  an xn + an1 xn1 + … + a2 x2 + a1 x + a0
where an  0. We say that,
The natural number n is the degree of the polynomial f.
The term anxn is the leading term of the polynomial f.
The real number an is the leading coefficient of the polynomial f.
The real number a0 is the constant term of the polynomial f.
If f (x)  a0, and a0  0, we say f has degree 0.
If f (x)  0, we say f has no degree. 4

5. Identifying Polynomial Functions
Determine which of the following functions are
polynomials. For those that are, state the degree.

6. Identifying Polynomial Functions
Determine which of the following functions are
polynomials. For those that are, state the degree. 6

7. Polynomial Functions: Example
A box with no top is to be built from a 10 inch by
12 inch piece of cardboard by cutting out congruent
squares from each corner of the cardboard and then
folding the resulting tabs.
Let x denote the length of the side of the square
which is removed from each corner.

8. Polynomial Functions: Example
A diagram representing the situation is,

9. Polynomial Functions: Example
1. Find the volume V of the box as a function of x. Include an appropriate applied domain.
2. Use a graphing calculator to graph y  V (x) on the domain you found in part 1 and approximate the dimensions of the box with maximum volume to two decimal places. What is the maximum volume?

10. Summary of the Properties of the Graphs of Polynomial Functions

11. Graphs of Polynomial Functions

12. Power Functions A power function of degree n is a function of the form
f (x)  axn
where a  0 is a real number and n  1 is an
integer.

13. Power Functions: a  1, n even

14. Power Functions: a  1, n even

15. Power Functions: a  1, n even

16. Power Functions: a  1, n odd

17. Power Functions: a  1, n odd

18. Power Functions: a  1, n odd

19. Identifying the Real Zeros of a Polynomial Function and Their Multiplicity

20. Graphs of Polynomial Functions

21. Definition: Real Zero

22. Finding a Polynomial Function from Its Zeros
Find a polynomial of degree 3 whose zeros are
 4,  2, and 3.
The value of the leading coefficient a is, at this point, arbitrary. The next slide shows the graph of three polynomial functions for different values of a.

23. Finding a Polynomial Function from Its Zeros23

24. Definition: Multiplicity
For the polynomial, list all zeros and their multiplicities.
2 is a zero of multiplicity 1 because the exponent on the factor x – 2 is 1.
1 is a zero of multiplicity 3 because the exponent on the factor x + 1 is 3.
3 is a zero of multiplicity 4 because the exponent on the factor x – 3 is 4.

25. Graphing a Polynomial Using Its x-Intercepts

28. Behavior Near a Zero

29. Example

30. Example
y = 4(x - 2)

31. y = 4(x - 2)

32. Turning Points: Theorem

33. End Behavior

34. End Behavior: Example 34

35. End Behavior: Example 35

39. Summary

40. Analyze the Graph of a Polynomial Function

43. The polynomial is degree 3 so the graph can turn at most 2 times.

45. Summary: Analyzing the Graph of a Polynomial Function

51. The domain and the range of f are the set of all real numbers.