C H A P T E R 2 Polynomial and Rational Functions

36. (a)
0 with multiplicity 2, 4 and 5 with multiplicity 1.
(b) The multiplicity of x 0 is 2.
(c)
38. (a)
fx x 4 x 3 20x 2 37.
x 0, 4, 5
The multiplicity of x 5 is 1.
The multiplicity of x 4 is 1.
There are three turning points.
−6
0 x 2 x 2 x 20
0 x 2 x 4x 5
25
−150
x 0, ±2, all with multiplicity 1
(b) The multiplicity of x 0 is 1.
(c)
40. (a)
6
f x x 5 x 3 6x 39.
The multiplicity of x 2 is 1.
The multiplicity of x 2 is 1.
There are two turning points.
−9
0 xx 4 x 2 6
0 xx 2 3x 2 2
6
−6
x ±5, both with multiplicity 1
(b) The multiplicity of x 5 is 1.
(c)
The multiplicity of x 5 is 1.
There is one turning point.
−6
20
−60
9
6
Section 2.2 Polynomial Functions of Higher Degree 157
gt t 5 6t 3 9t
(a)
Zeros:
(b) t 0 has a multiplicity of 1 (odd multiplicity).
(c)
(a)
0 t 5 6t 3 9t tt 4 6t 2 9 tt 2 3 2
t ±3 each have a multiplicity of 2
(even multiplicity).
Turning points: 4
−9
tt 3 2
t 3 2
t 0, t ±3
6
−6
fx 5x 4 15x 2 10
No real zeros
(b) Turning point: 1
(c)
0 5x 4 15x 2 10
−4
5x 4 3x 2 2
5x 2 1x 2 2
f x 2x4 2x2 40 41. gx x3 3x2 4x 12
0 2x 4 2x 2 40
0 2x 2 4x 5x 5
(a)
Zeros:
40
−5
9
4
(b) Each zero has a multiplicity of 1 (odd multiplicity).
(c)
0 x 3 3x 2 4x 12 x 2 x 3 4x 3
Turning points: 2
−8
x 2 4x 3 x 2x 2x 3
x ± 2, x 3
4
−16
7