Math 113
Lecture Notes 10: Polynomials 3.2
3
Numerical Example
Consider the polynomial
.
x
1
1
3
4
6
10
100,000
3,000
40
96,960
100
10,000,000,000
3,000,000
400
9,996,999,600
1,000
1E+15 = _________
3,000,000,000
4,000
10,000
1E+20 =__________
40,000
as
x
®

x
5
®
3
x
3
®
4
x
®
f
(
x
)
®
In this example, which term dominates? _____________.
What is the end behavior displayed here?
In a nutshell
The end behavior of the polynomial
is determined by the
degree
n
and the sign of the leading coefficient
a
n
.
P
(
x
) has odd degree (
n
is odd)
Leading coefficient is positive (
a
n
>
0)
Leading coefficient is negative (
a
n
<
0)
Example formula:
Example formula:
End behavior:
as
as
End behavior:
as
as
f
(
x
)
=
−
x
5
+
3
x
3
+
4
x
−
x
5
3
x
3
4
x
f
(
x
)
=
−
x
5
+
3
x
3
9.99997
×
10
15
3
×
10
12
−
1
×
10
20
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
y
→
________
x
→
+
∞
y
→
________
x
→ −
∞
y
→
________
x
→
+
∞
y
→
________
x
→ −
∞