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Characteristics of
Polynomial Functions
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Many real-life situations can be modeled by polynomial functions.
For example, the volume of a silo is a polynomial function of its
radius.
In this lesson you will investigate the following properties of
polynomial functions :
 A polynomial function is an equation of the
form f(x)=a
n
x
n
+a
n−1
x
n−1
+...+a
2
x
2
+a
1
x+a
0
f(x)=anxn+an-1xn-
1+...+a2x2+a1x+a0, where the coefficients a
n
an, a
n−1
an-
1, ..., a
1
a1, a
0
a0 represent real numbers, a
n
an is not zero, and
the exponents are non-negative integers.
 The real zeros of a function are the xx-intercepts of its graph.
 You can describe the behavior of a function based on its
degree, the leading coefficient and the constant term.
Review
Before beginning this lesson you should be able to:
 Use a graphing calculator to find the x and y -intercepts.

 Analyze quadratic functions to identify characteristics of the
corresponding graph, including:
o vertex
o domain and range
o direction of opening
o axis of symmetry
o xx- and yy-intercepts
Polynomial Characteristics Video
Note: The video below covers the concepts from the lesson but is
different than the examples that follows.
Examples and non-examples of polynomial functions include:
Examples Non Examples
f(x)=5x
3
−8x
2
+x+1 f(x)=5x3-8x2+x+1 f(x)=xx
2
−3 f(x)=xx2-3
g(x)=6x
3
g(x)=6x3
g(x)=2
√ x
−4x+5 g(x)=2x
-4x+5
h(x)=2.1x
4
+3.5x
3
−4x
2
+x−7 h(x)=2.1x4+3.5
x3-4x2+x-7
h(x)=√x−4 h(x)=x-4
Notice that a function CANNOT be classified as a polynomial function
if:
 an exponent is a fraction, negative number, radical, etc. All
exponents must be a positive integers.

 a variable is in the denominator. All variables must exist in the
numerators only.
 a variable is located within a root sign
Let's look at a specific example to cover some important terms:
Example 1
f(x)=2x
4
+3x
3
−x
2
−5x+10f(x)=2x4+3x3-x2-5x+10.
 The coefficients are 2, 3, -1, -5 and 10. They can be any real
number.
 The exponents are 4, 3, 2 and 1. They must be positive
integers.
 The degree of the polynomial is 4. This is the highest
exponent on a variable. The terms should always be arranged
such that the exponents are in descending order.
 The leading coefficient is 2. It is the coefficient of the
highest degree term.