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:
Pre

Calculus

Chapter 5A
Chapter 5A

Polynomial Functions
Definition of Polynomial Functions
A
polynomial function
is any function
p
x
of the form
p
x
p
n
x
n
p
n
−
1
x
n
−
1
p
2
x
2
p
1
x
p
0
,
where all of the exponents are nonnegative integers.
The coefficients of the various powers of
x
,thatis
,the
p
i
’s are assumed to be known real
numbers, and
p
n
≠
0.
The degree of this polynomial function is
n
.If
n
is even the
polynomial is said to be of even degree and if
n
is odd, the polynomial is said to be of odd
degree.
The coefficient
p
0
is called the
constant term
and the nonzero
p
n
is called the
leading
coefficient
.
By a polynomial of degree 0, we mean a nonzero constant function. The zero polynomial
function is sometimes said to have degree equal to
−
.
In the table below we list a few polynomials and their degrees.
polynomial
degree constant term leading coefficient
p
x
50
5
5
p
x
3
x
2
−
82
−
83
p
x
−
6
x
5
x
4
−
5
x
95
9
−
6
p
x
2
x
126
126
0
2
We have studied two examples of polynomial functions already. Linear functions (
f
x
mx
b
,
m
≠
0) are polynomial functions of degree 1, and quadratic functions (
f
x
ax
2
bx
c
,
a
≠
0) are
polynomial functions of degree 2.
Below are some examples of polynomials of degree 3, 4, and 5.
Example 1
:L
e
t
p
x
5
−
6
x
13
x
3
. What is the degree of
p
x
? What are the leading coefficient
and constant terms?
Solution:
The degree of
p
x
5
−
6
x
−
13
x
3
is 3. The largest exponent. The leading coefficient
is
−
13 and the constant term is 5.
Example 2
e
t
p
x
3
x
4
−
3
x
3
5
x
. What is the degree of
p
x
? What are the leading coefficient
and constant terms?
Solution:
The degree of
p
x
3
x
4
−
3
x
3
5
x
is 4. The leading coefficient is 3, and the constant
term, the coefficient of
x
0
,is0
.
Example 3
:I
f
f
x
16
−
3
x
2
27
x
4
−
5
x
7
8
x
13
, what are the degree, leading coefficient, and
constant term equal to?
Solution:
The degree of
f
x
16
−
3
x
2
27
x
4
−
5
x
7
8
x
13
is 13. The leading coefficient is 8,
and the constant term is 16.
©
:
Pre

Calculus
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Chapter 5A
Behavior of Polynomial Functions
In this page we will discuss how the values
p
x
of polynomial functions behave for large values of
x
,
both positive and negative. It turns out that the behavior of polynomial functions for large values of
x
is determined by the degree of the polynomial. Polynomials of odd degree behave one way, think of the
graph of a linear function, and polynomials of even degree behave in a different way, picture the graph
of a quadratic function.
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 Fall '07
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