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Polynomial and Rational Functions
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4.2
Polynomial Functions
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3. Objectives Recognize Polynomial Functions
Understand Characteristics of the Graphs of Polynomial Functions
Find Zeros of Polynomial Functions by Factoring
Determine End Behavior

4. Objectives Graph Polynomial Functions
Use the Intermediate Value Theorem

5. 1. Recognize Polynomial Functions

6. Polynomial Functions
A polynomial function in one variable (say, x) is a function of the form
f(x) = anxn + an–1xn–1 + … + a1x + a0
where an, an–1, … , a1,and a0 are real numbers and n is a whole number.
The degree of a polynomial function is the largest power of x that appears in the polynomial.
The real number an is referred to as the leading coefficient.

7. Examples
The table shows examples of polynomial functions and examples of functions that are not polynomials.

8. Example 1
Determine whether or not the functions are a polynomial. For those that are, state the degree. a. b. c.

9. Example 1(a) – Solution is a polynomial function of degree 6.
Note that each exponent on the variable x is a whole number and that 6 is the largest power of x.

10. Example 1(b) – Solution is not a polynomial function.
Note that even though each exponent on the variable x is a whole number, x appears in the denominator.
For that reason, the definition of a polynomial function is not satisfied.

11. Example 1(c) – Solution is not a polynomial function.
Note that the exponents –4 and –2 are not whole numbers.

12. 2. Understand the Characteristics of the Graphs of Polynomial Functions

13. Characteristics of Polynomial Functions
There are several characteristics of the graphs of polynomial functions that should be noted.
We will list five of them.

14. Characteristic 1
The graphs of polynomial function is a smooth and continuous curve.
Like the graphs of linear and quadratic functions, the graphs of higher-degree polynomial functions are smooth continuous curves.
Because their graphs are smooth, they have no cusps or corners.
Because they are continuous, their graphs have no breaks or holes.
They always can be drawn without lifting the pencil from the paper.

15. Characteristic 1
The graph of a polynomial function

16. Characteristic 1
The graphs of two functions that are not polynomial functions

17. Characteristic 2
The graph of a polynomial function has a specific end behavior and will either rise or fall on the far left and far right.
By end behavior, we mean what happens on the far left and far right on the graph.
A polynomial function’s graph will either rise or fall on its ends.

18. Characteristic 2
Note that the graph of the polynomial function in first Figure falls on the left and rises on the right.
Note that the polynomial function in second Figure rises on both the left and right.

19. Characteristic 3
The graph of a polynomial function often has x-intercepts.
The x-values of the x-intercepts are known as zeros of the polynomial function.
A zero of a polynomial function is a value of x for which f (x) = 0.
It is interesting to note that the graph of a polynomial function will either touch and turn at the zero of the polynomial function or cross the x-axis there.

20. Characteristic 4
The graph of a polynomial function often has turning points.
Turning points on the graph are where maxima and minima occur.
Finding the exact turning points often requires calculus techniques.

21. Characteristic 5
The graph of a polynomial functions can be symmetric about the y-axis or the origin.
Function’s graph is symmetric about the y-axis if f (–x) = f (x).
A function’s graph is symmetric about the origin if f(–x) = –f(x).

22. Characteristic 5
Symmetry can be very helpful when graphing a polynomial function.

23. 3. Find Zeros of Polynomial Functions by Factoring

24. Zeros of Polynomial Functions
We used factoring to solve the polynomial equations.
If f (x) represents a polynomial function, then the solutions or roots of the polynomial equation f (x) = 0 are known as zeros of the polynomial function.

25. Zeros of Polynomial Functions
In this section, we will find zeros of polynomial functions which can be factored.
However, not all polynomials can be factored using the factoring techniques.
Each zero of the polynomial function appears as an x-intercept on the graph of the polynomial function.

26. Example 2
Find the zeros of polynomial function.

27. Example 2 – Solution
We set f (x) = 0 and factor by grouping to find the zeros.

28. Example 2 – Solution
The zeros of the polynomial function are x = –4, x = –1, and x = 1. The graph of the polynomial function has x-intercepts (–4, 0), (–1, 0), and (1, 0) as shown in Figure.

29. 4. Determine End Behavior

30. Power Function
A basic polynomial function, called a power function, is a polynomial function of the form f (x) = axn, where a is a real number, a ≠ 0, and n is a nonnegative integer.
Power functions of the form f (x) = xn have simple graphs.

31. Graphs of Power Function

32. Leading Coefficient Test
The end behavior of the graph of a polynomial function is the same as the graph of its term with highest degree.
The leading coefficient, an, and the degree, n, of a polynomial will be our starting point.
These will determine the end behavior. This is summarized in the Leading Coefficient Test.

33. Coefficient Test and End Behavior
For as x increases without bound or decreases without bound , the function will eventually increase without bound or decrease without bound

34. Summary of Cases
Case 1: If the degree of the polynomial is odd and the leading coefficient is positive, then the graph of the polynomial function falls on the left and rises on the right.

35. Summary of Cases
Case 2: If the degree of the polynomial is odd and the leading coefficient is negative, then the graph of the polynomial function rises on the left and falls on the right.

36. Summary of Cases
Case 3: If the degree of the polynomial is even and the leading coefficient is positive, then the graph of the polynomial function rises on the left and rises on the right.

37. Summary of Cases
Case 4: If the degree of the polynomial is even and the leading coefficient is negative, then the graph of the polynomial function falls on the left and falls on the right.

38. Example 3
Use the Leading Coefficient Test to describe the end behavior of function. a. b. c. d.

39. Example 3(a) – Solution
The degree of is 3, which is odd, and the leading coefficient is 4, which is positive.
Case 1 applies. The end behavior of the function will be like that of f (x) = x3.
The graph falls on the left and rises on the right.

40. Example 3(b) – Solution
The degree of is 5, which is odd, and the leading coefficient is –2, which is negative.
Case 2 applies. The end behavior of the function will be like that of f (x) = –x3.
The graph rises on the left and falls on the right.

41. Example 3(c) – Solution
The degree of is 4, which is even, and the leading coefficient is 5, which is positive.
Case 3 applies. The end behavior of the function will be like that of f (x) = x2.
The graph rises on the left and rises on the right.

42. Example 3(d) – Solution
The degree of is 6, which is even, and the leading coefficient is –7, which is negative.
Case 4 applies. The end behavior of the function will be like that of f (x) = –x2.
The graph falls on the left and falls on the right.

43. 5. Graph Polynomial Functions

44. Multiplicity and Zeros
If a is a zero with an odd multiplicity, the graph crosses the x-axis at x = a.
If a is a zero with an even multiplicity, the graph touches the x-axis and turns at x = a.

45. Number of Turning Points
If f (x) is a polynomial function of degree n, then the graph of f (x) will have n – 1, or fewer, turning points.

46. Strategy for Graphing Functions
Find the x- and y-intercepts of the graph.
2. Determine the end behavior.
3. Make a sign chart and determine where the graph is above and below the x-axis.
4. Find any symmetries of the graph.
5. Plot a few points, if necessary, and draw the graph as a smooth, continuous curve.

47. Example 4
Graph the function:
f(x) = x3 – 4x

48. Example 4 – Step 1 Find the x- and y-intercepts of the graph.
To find the x-intercepts, we let f(x) = 0 and solve for x.

49. Example 4 – Step 1 The x-intercepts are (0, 0), (–2, 0), and (2, 0).
If we let x = 0 and solve for f(x), we see that the y-intercept is also (0, 0).

50. Example 4 – Step 2 Determine the end behavior.
The degree of the function f (x) = x3 – 4x is 3, which is odd, and the leading coefficient is 1, which is positive.
The end behavior of the function is like that of the function f (x) = x3. The graph falls on the left and rises on the right.

51. Example 4 – Step 3
Make a sign chart to determine where the graph is above and below the x-axis.
We will make a sign chart to determine the behavior of the function between the x-intercepts.
The zeros of the polynomial function divide the x-axis into four intervals:

52. Example 4 – Step 3
We then test a number from each interval to determine the sign of f(x).

53. Example 4 – Step 4 Find any symmetries of the graph.
Since f(x) ≠ f(–x), there is no symmetry about the y-axis.
However, since f(–x) = –f(x), there is symmetry about the origin.

54. Example 4 – Step 5
Plot a few points and draw the graph as a smooth continuous curve.

55. 6. Use the Intermediate Value Theorem

56. Intermediate Value Theorem
Let P(x) be a polynomial function with real coefficients.
If P(a) ≠ P(b) for a < b, then P(x) takes on all values between P(a) and P(b) on the closed interval [a, b].

57. Example 6
Show that P (x) = 2x3 – x2 – 8x + 4 has at least one real zero between 0 and 1.

58. Example 6 – Solution We will evaluate the polynomial function at
x = 0 and x = 1.
If the resulting values have opposite signs, we know that the equation has a zero that lies between 0 and 1.

59. Example 6 – Solution
Evaluate P(0) and P(1) as follows:

60. Example 6 – Solution
Because P(0) and P(1) have opposite signs, we know that there is at least one real zero between 0 and 1, as shown in Figure.
The zero shown is 1/2.