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

194 Chapter 2 Polynomial and Rational Functions
51. gx 4x
Since 3 i is a zero, so is 3 i.
3 23x2 34x 10
3 i 4
4
3 i 4
4
23
12 4i
11 4i
11 4i
12 4i
1
34
37 i
3 i
3 i
3 i
0
10
10
0
The zero of The zeros of are
x 3 ± i and x 1
4 .
4x 1 is x gx
1
4 .
52. hx 3x3 4x 2 8x 8
53.
Since 1 3 i is a zero, so is 1 3i.
1 3i 3
3
1 3i 3
3
4
3 33i
1 33i
1 33i
3 33i
2
The zero of The zeros of f are x 2
3x 2 is x 2
, 1 ± 3i.
fx x 4 3x 3 5x 2 21x 22
3 .
8
10 23i
2 23i
2 23i
2 23i
Since 3 2 i is a zero, so is 3 2 i, and
is a factor of fx. By long division, we have:
x 2 6x 11 ) x 4 3x 3 5x 2 21x 22
Thus,
x 4 6x 3 11x 2
3x 3 16x 2 21x
3x 3 18x 2 33x
f x x 2 6x 11x 2 3x 2
x 2 6x 11x 1x 2
0
x 2 3x 2
2x 2 12x 22
2x 2 12x 22
8
8
x 3 2 i x 3 2 i x 3 2 ix 3 2 i
and the zeros of f are x 3 ± 2 i, x 1, and x 2.
0
0
x 3 2 2 i 2
x 2 6x 11
3
Alternate Solution
Since 3 ± i
are zeros of gx,
x 32 i2 x 3 i x 3 i x 3 ix 3 i
x
is a factor of gx. By long division we have:
2 6x 10
34x 1
x 2 6x 10 ) 4x 3 23x 2 34x 10
4x 3 24x 2 40x
4x 3 24 x 2 36x 10
x 2 36x 10
Thus, and the zeros of
are x 3 ± i and x 1
gx x gx
4.
2 6x 104x 1
0