1. Polynomial Functions and End Behavior
On to Section 2.3!!!

2. Definitions: The Vocabulary of Polynomials
Cubic Functions – polynomials of degree 3
Quartic Functions – polynomials of degree 4
Recall that a polynomial function of degree n can be written
in the form:

3. Definitions: The Vocabulary of Polynomials
Each monomial is this sum is a
term of the polynomial.
A polynomial function written in this way, with terms in
descending degree, is written in standard form.
The constants are the coefficients of the
polynomial.
The term is the leading term, and is the constant
term.

4. Practice Problems
Graph the polynomial function, locate its extrema and
zeros, and explain how it is related to the monomials
from which it is built.
Graph with your grapher!!!
Increasing on entire domain  No extrema!!!
One zero  At x = 0!!!
General shape of the graph is much like that of the leading
term (cubing function), but near the origin behaves much
like the its other term (identity function)
Function is odd, just like its two “building block” monomials

5. Practice Problems
Graph the polynomial function, locate its extrema and
zeros, and explain how it is related to the monomials
from which it is built.
Graph with your grapher!!!
Local maximum of approx at about x = –0.577
Local minimum of approx. –0.385 at about x = 0.577
Zeros at x = –1, 0, 1 (factored version of the function?)
Graph is generally similar to leading term, but near the
origin behaves like the other term, –x
Function is odd, just like its two “building block” monomials

6. Summarize these thoughts…
Theorem: Local Extrema and Zeros of
Polynomial Functions
A polynomial function of degree n has at most
n – 1 local extrema and at most n zeros.
Does this make sense with the
graphs we were just looking at???

7. Leading Term Test for Polynomial End Behavior
For any polynomial function, the end behavior is determined by
its degree and leading coefficient:
n odd
n odd

8. Leading Term Test for Polynomial End Behavior
For any polynomial function, the end behavior is determined by
its degree and leading coefficient:
n even
n even

9. Practice Problems One possible window: [–5, 5] by [–25, 25]
Graph the polynomial function in a window showing its
extrema and zeros and its end behavior. Describe the
end behavior using limits.
n odd
Graph with your grapher!!!
One possible window: [–5, 5] by [–25, 25]
Function has 2 extrema and 3 zeros

10. One possible window: [–5, 5] by [–50, 50]
Practice Problems
Graph the polynomial function in a window showing its
extrema and zeros and its end behavior. Describe the
end behavior using limits.
n even
Graph with your grapher!!!
One possible window: [–5, 5] by [–50, 50]
Function has 3 extrema and 4 zeros

11. Zeros of Polynomial Functions
Recall that finding the real-number zeros
of a function f is equivalent to finding the
x-intercepts of the graph of y = f (x) or the
solutions to the equation f (x) = 0.

12. A Quick Example… Can we support our answer graphically??? Zero Factor
Find the zeros of
We solve the equation f (x) = 0 by factoring:
Can we support
our answer
graphically???
Zero Factor
Property!!!
x = 0, x – 3 = 0, or x + 2 = 0
Zeros of f : 0, 3, –2

13. Definition: Multiplicity of a Zero of a Polynomial Function
If f is a polynomial function and is
a factor of f but is not, then c is a
zero of multiplicity m of f.
A zero of multiplicity m > 2 is a repeated zero.

14. Zeros of Odd and Even Multiplicity
If a polynomial function f has a real zero c of odd
multiplicity, then the graph of f crosses the x-axis
at (c, 0) and the value of f changes sign at x = c.
If a polynomial function f has a real zero c of even
multiplicity, then the graph of f does not cross the
x-axis at (c, 0) and the value of f does not change
sign at x = c.

15. Examples with our new info…
Consider the function
What are the zeros of the function?
2 and –1
What are the multiplicity of these zeros?
2 is a zero of multiplicity 3, –1 has multiplicity 2
What does the graph of the function look like at these zeros?
At x = 2, the graph crosses the x-axis,
and at x = –1, the graph “kisses” the x-axis
Verify all of this with your calculator!!!

16. Examples with our new info…
State the degree and list the zeros of the given function.
State the multiplicity of each zero and whether the graph
crosses the x-axis at the corresponding x-intercept. Then
sketch the graph of the function by hand.
Degree is 5, zeros are x = –2 (multiplicity 3) and
x = 1 (multiplicity 2). The graph crosses the x-axis
at x = –2, but not at x = 1.

17. Intermediate Value Theorem
Essentially, a sign change implies a real zero…
If a and b are real numbers with a < b and if f is continuous
on the interval [a, b], then f takes on every value between
f(a) and f(b). In other words, if y is between f(a) and f(b),
then y = f(c) for some number c in [a, b].
In particular, if f(a) and f(b) have opposite signs (i.e., one is
negative and the other is positive), then f(c) = 0 for some
number c in [a, b].

18. Intermediate Value Theorem
Let’s “see” this theorem graphically:
If f (a) < 0 < f (b), then there is
a zero x = c between a and b

19. Guided Practice Zeros at x = –3.100, 0.5, 1.133, and 1.367
Find all of the real zeros of the given function.
Because the function has degree 4, there are at most four
zeros  Graph and find them!!!
Zeros at x = –3.100, 0.5, 1.133, and 1.367
Homework: p odd, odd