Section 3.3 Polynomial Functions
Polynomial of degree
n
where
n
is a nonnegative integer and
a
n
6
= 0:
f
(
x
) =
a
n
x
n
+
a
n

1
x
n

1
+
· · ·
+
a
1
x
+
a
0
Example.
Polynomial function:
f
(
x
) =

2
x
3
+ 4
x
Example.
Polynomial function:
f
(
x
) = 4
Example.
NOT a polynomial function:
f
(
x
) =
x

3
+
x
Example.
NOT a polynomial function:
f
(
x
) =
√
x
Graphs of polynomial functions are
•
continuous (no breaks)
•
smooth (no sharp corners)
Graph of
y
= (
x
+ 1)(
x
)(
x

1)(
x

2):
–2
–1
0
1
2
–2
–1
1
2
Finding a Polynomial with given Zeros:
Example.
Find a polynomial of degree 3, with zeros at 3, 2, 1.
Solution.
f
(
x
) =
a
(
x

3)(
x

(

2))(
x

1) =
a
(
x

3)(
x
+ 2)(
x

1), with
a
6
= 0 arbitrary. The
graph of
f
is shown below with
a
= 1.
–12
–6
0
6
12
–3
–2
–1
1
2
3
4
y
= (
x

3)(
x
+ 2)(
x

1)
Example.
Find a polynomial with zeros of 3, 2, 1 having degree 4 and with 3 a zero of multiplicity
two.
Solution.
f
(
x
) =
a
(
x

3)
2
(
x

(

2))(
x

1) =
a
(
x

3)
2
(
x
+ 2)(
x

1), with
a
6
= 0 arbitrary.
1
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–30
–20
–10
10
–3
–1
1
3
y
= (
x

3)
2
(
x
+ 2)(
x

1)
The role of multiplicity in graphing polynomials:
The graphs of
f
(
x
) =
a
(
x

1)(
x

2)
n
(
x

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 Fall '11
 Kutter
 Algebra, polynomial function

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