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Section 5.1
Polynomial Functions and
Models
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.
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 ( , ).
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.
Identifying Polynomial Functions
Determine which of the following functions are
polynomials. For those that are, state the degree.
(a) f  x   3x  4x3  x8
x2  3
(b) g  x  
x 1
(a) f is a polynomial of degree 8.
(b) g is not a polynomial function.
It is the ratio of two distinct polynomials.
(c) h  x   5
(d) F  x   ( x  3)( x  2)
(c) h is a polynomial function of degree 0.
(d) F is a polynomial function of degree 2.
It can be written h  x   5 x 0  5.
It can be written F ( x)  x 2  x  6.
Identifying Polynomial Functions
Determine which of the following functions are
polynomials. For those that are, state the degree.
(e) G  x   3x  4x1
(e) G is not a polynomial function.
The second term does not have a
nonnegative integer exponent.
(f) H  x  
1 3 2 2 1
x  x  x
2
3
4
(f) H is a polynomial of degree 3.
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.
Polynomial Functions: Example
A diagram representing the situation is,
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?
Summary of the Properties of the
Graphs of Polynomial Functions
Graphs of Polynomial Functions
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.
Power Functions: a  1, n even
Power Functions: a  1, n even
Power Functions: a  1, n even
Power Functions: a  1, n odd
Power Functions: a  1, n odd
Power Functions: a  1, n odd
Identifying the Real Zeros of a
Polynomial Function and
Their Multiplicity
Graphs of Polynomial Functions
Definition: Real Zero
Finding a Polynomial Function
from Its Zeros
Find a polynomial of degree 3 whose zeros are
 4,  2, and 3.
f  x   a  x  4 x  2 x  3  a  x3  3x 2  10 x  24 
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.
Finding a Polynomial Function
from Its Zeros
f  x    x  4 x  2 x  3
f  x   2  x  4 x  2 x  3
f  x     x  4 x  2 x  3
Definition: Multiplicity
For the polynomial, list all zeros and their multiplicities.
f  x   2  x  2  x  1  x  3
3
4
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.
Graphing a Polynomial Using
Its x-Intercepts
f  x   x  x  3
(a) x-intercepts: 0  x  x  3
2
2
x  0 or  x  3  0
2
x  0 or x  3
y -intercept: f  0   0  0  3  0
2
y0
f  x   x  x  3
2
 0,0 , 3,0
 , 0
 0,3
3,
1
1
4
f  1  16
f 1  4
f  4  4
Below x-axis
Above x-axis
Above x-axis
 1, 16
1, 4
 4, 4 
y



f  x   x  x  3

2
x













 , 0
 0,3
3,
1
1
4
f  1  16
f 1  4
f  4  4
Below x-axis
Above x-axis
Above x-axis
 1, 16
1, 4
 4, 4 
Behavior Near a Zero
Example
Example
y = 4(x - 2)
y = 4(x - 2)
Turning Points: Theorem
End Behavior
End Behavior: Example
End Behavior: Example
f  0  6 so the y intercept is  6.
The degree is 4 so the graph can turn at most 3 times.
For large values of x, end behavior is like x4 (both ends approach )
Summary
Analyze the Graph of a
Polynomial Function
1
has multiplicity 1
2
so the graph crosses there.
The zero 
The zero 3 has multiplicity 2
so the graph touches there.
The polynomial is degree 3 so the
graph can turn at most 2 times.
Summary: Analyzing the Graph
of a Polynomial Function
The domain and the range of f are
the set of all real numbers.
Decreasing:
Increasing:
 2.28, 0.63
 , 2.28 and  0.63,  