SECTION 41
211
CHAPTER 4
Polynomial and Rational Functions
Section 41
1.
The degree is odd and the coefficient is positive, so
f
(
x
) increases without
bound as
±
±
and decreases without bound as
±

±
. This matches graph c.
3.
The degree is even and the coefficient is positive, so
h
(
) increases without
bound both as
±
±
and
±

±
. This matches graph d.
5.
The real zeros are the
intercepts: 1 and 3. The turning point is (1, 4).
P
(
)
±

±
as
±
±
and
(
)
±

±
as
±

±
.
7.
The real zeros are the
intercepts: 2 and 1. The turning points are (1, 4)
and (1, 0).
(
)
±

±
as
±

±
and
(
)
±
±
as
±
±
.
9.
The graph of a polynomial always increases or decreases without bound as
±
±
and
±

±
. This graph does not.
11.
The graph of a polynomial always has a finite number of turning pointsat most
one less than the degree. This graph has infinitely many turning points.
13.
Set each factor equal to zero:
= 0
2
 9 = 0
2
+ 4 = 0
2
= 9
2
= 4
= 3, 3
= 2
i
, 2
The zeros are 0, 3, 3, 2
, and 2
; of these 0, 3, and 3 are
intercepts.
15.
Set each factor equal to zero:
+ 5 = 0
2
+ 9 = 0
2
+ 16 = 0
= 5
2
= 9
2
= 16
= 3
, 3
= 4
, 4
The zeros are 5, 3
, 3
, 4
, and 4
. Only 5 is an
intercept.
17.
3
+ 2
+
13
2
+
5
+
6
3
2
+
3
2
+
6
2
+
2
4
3
+ 2,
R
= 4
19.
2
m
+ 1
2
 1 4
2
+ 0
 1
4
2
 2
2
 1
2
 1
0
2
+ 1,
= 0
21.
4
 5
2
+ 1 8
2

6
+
6
8
2
+
4
 10
+
6
 10

5
11
4
 5,
= 11
23.
2
+
+ 1
 1
3
 0
2
+ 0
 1
3

2
2
+ 0
2

 1
 1
0
2
+
+ 1,
= 0