“Why so serious?”
3.2 Polynomial Functions and Models
Polynomial Functions
A polynomial function is a function of the form
f (x)  an x  an1 x  ... a1 x  a0
where an ,an1,...,a1,a0 are real numbers and n is a
n
n1
nonnegative integer. The domain consists of all real numbers.


The degree of the function is the largest power of x.
Polynomial Functions
Determine which of the following are polynomial functions.
For those that are, state the degree.
f (x)  3  2x 5
g(x)  x  5
h(x)  3x 3 (x  2)2
3x 5
F(x) 
5  2x
G(x)  6
Polynomial Functions
Determine which of the following are polynomial functions.
For those that are, state the degree.
4 x 3  3x  5
f (x) 
2
g(x)  7x(2x  3) 4 (x  5) 2
h(x)  x  ex  2.634
3
2
F(x)  x  2x  8
4
G(x)  x  2x  3
e
Power Functions
A power function is of the form
f (x)  x
If n is odd…
n
If n is even…

The end behavior of a polynomial function can be related to
the power function of the same degree.
Zeroes of Polynomials
Find the degree, the end behavior, zeroes, and multiplicity each.
f (x)  x 2 (x  3)(x  2) 3

If a zero has an odd multiplicity, the function crosses at the zero.
If a zero has an even multiplicity, the function touches at the zero.
Zeroes of Polynomials
Find the degree, the end behavior, zeroes, and multiplicity each.
f ( x)   x3 ( x  2) 4 ( x  8)
Zeroes of Polynomials
Find the polynomial whose zeroes are given.
1) Zeroes: 0, -4, 2 (all multiplicity 1)
2) Zeroes: -3, multiplicity 2; 5, multiplicity 1
3) Zeroes: 0, multiplicity 2; 6, multiplicity 3; -2, multiplicity 4
Zeroes of Polynomials
Given the polynomial f (x)  x(x  4) 2 (x  3)(x 2  1)
a) Find the degree and end behavior of the polynomial.
b) Find the x-and y-intercepts of the graph.
c) Determine whether the graph crosses or touches at each xintercept.
d) Graph f using a graphing utility.
e) Determine any local maxima or minima.
Zeroes of Polynomials
Given the polynomial f (x)  x(x  4) 2 (x  3)(x 2  1)
f) Graph by hand.
g) Determine the domain and range.

h) Determine intervals of increasing
and decreasing
Degree = 6 y-intercept: (0,0)
x-intercepts: (0,0) crosses, (4,0) touches; (-3,0) crosses
Local minima: (-2.321,-402.176) and (4,0)
Local maxima: (2.451,224.636)
3.2 Polynomial Functions and Models