1. POLYNOMIAL FUNCTIONS AND MODELS
SECTION 3.1
POLYNOMIAL FUNCTIONS AND MODELS

2. f(x) = a n x n + a n-1 x n-1 + . . . + a1x + a0
POLYNOMIAL FUNCTIONS
A polynomial is a function of the form
f(x) = a n x n + a n-1 x n a1x + a0
where an, a n-1, . . ., a1, a0 are real numbers and n is a nonnegative integer.
The domain consists of all real numbers.

3. POLYNOMIAL FUNCTIONS
Which of the following are polynomial functions?

4. POLYNOMIAL FUNCTIONS
SEE TABLE 1

5. POLYNOMIAL FUNCTIONS
The graph of every polynomial function is smooth and continuous: no sharp corners and no gaps or holes.

6. POLYNOMIAL FUNCTIONS
When a polynomial function is factored completely, it is easy to solve the equation f(x) = 0 and locate the x-intercepts of the graph.
Example: f(x) = (x - 1)2 (x + 3) = 0
The zeros are 1 and - 3

7. POLYNOMIAL FUNCTIONS
If f is a polynomial function and r is a real number for which f (r ) = 0, then r is called a (real) zero of f , or root of f.
If r is a (real) zero of f , then
(a) r is an x-intercept of the graph of f.
(b) (x - r) is a factor of f.

8. POLYNOMIAL FUNCTIONS
If (x - r)m is a factor of a polynomial f and (x - r)m+1 is not a factor of f, then r is called a zero of multiplicity m of f.
Example: f(x) = (x - 1)2 (x + 3) = 0
1 is a zero of multiplicity 2.

9. POLYNOMIAL FUNCTIONS
For the polynomial
f(x) = 5(x - 2)(x + 3)2(x - 1/2)4
2 is a zero of multiplicity 1
- 3 is a zero of multiplicity 2
1/2 is a zero of multiplicity 4

10. INVESTIGATING THE ROLE OF MULTIPLICITY
For the polynomial f(x) = x2(x - 2)
(a) Find the x- and y-intercepts of the graph.
(b) Graph the polynomial on your calculator.
(c) For each x-intercept, determine whether it is of odd or even multiplicity.
What happens at an x-intercept of odd multiplicity vs. even multiplicity?

11. EVEN MULTIPLICITY
If r is of even multiplicity:
The sign of f(x) does not change from one side to the other side of r.
The graph touches the x-axis at r.

12. ODD MULTIPLICITY
If r is of odd multiplicity:
The sign of f(x) changes from one side to the other side of r.
The graph crosses the x-axis at r.

13. TURNING POINTS
When the graph of a polynomial function changes from a decreasing interval to an increasing interval (or vice versa), the point at the change is called a local minima (or local maxima). We call these points TURNING POINTS.

14. EXAMPLE
Look at the graph of f(x) = x3 - 2x2
How many turning points do you see?
Now graph:
y = x3, y = x3 - x,
y = x3 + 3x2 + 4

15. EXAMPLE
Now graph:
y = x4, y = x4 - (4/3)x3,
y = x4 - 2x2
How many turning points do you see on these graphs?

16. THEOREM
If f is a polynomial function of degree n, then f has at most n - 1 turning points.
In fact, the number of turning points is either exactly n - 1or less than this by a multiple of 2.

17. GRAPH:
P(x ) = x2 P2(x) = x3
P1(x) = x4 P3(x) = x5

18. When n (or the exponent) is even, the graph on both ends goes to ± ¥.
When n is odd, the graph goes in opposite directions on each end, one toward + ¥ , the other toward - ¥.

19. EXAMPLE:
Determine the direction the arms of the graph should point. Then, confirm your answer by graphing.
f(x) = x 7

20. EXAMPLE:
Graph the functions below in the same plane, first using [- 10,10] by [- 1000, 1000], then using [- 10, 10] by [ , 10000]:
p(x) = x 5 - x x x + 3
p(x) = x 5

21. The behavior of the graph of a polynomial as x gets large is similar to that of the graph of the leading term.

22. f(x) = a n x n + a n-1 x n-1 + . . . + a1x + a0
THEOREM
For large values of x, either positive or negative, the graph of the polynomial
f(x) = a n x n + a n-1 x n a1x + a0
resembles the graph of the power function
y = a n x n

23. EXAMPLE
DO EXAMPLES 9 AND 10

24. CONCLUSION OF SECTION 3.1