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

47.
48.
fx 2x 3 3x 2 50x 75
Since 5i is a zero, so is 5i.
5i 2
2
5i 2
The zero of The zeros of
are x 3
2x 3 is x fx
2 and x ±5i.
3
2 .
49. fx 2x
Since 2i is a zero, so is 2i.
4 x3 7x2 4x 4
2i 2
2
2
2i 2
2
3
10i
3 10i
3 10i
10i
3
1
4i
1 4i
1 4i
4i
1
50
50 15i
15i
15i
15i
0
f x x 3 x 2 9x 9
Since 3i is a zero, so is 3i.
3i 1
1
3i 1
1
1
3i
1 3i
1 3i
3i
1
9
9 3i
3i
3i
7
8 2i
1 2i
1 2i
2i
1
75
75
0
4
4 2i
2i
2i
2i
0
The zeros of
are and The zeros of are
x ±2i, x and x 1.
1
2 ,
x x 1.
fx
1
2x
2
2 x 1 2x 1x 1
0
3i
9
9
0
4
4
0
Alternate Solution
The zero of x 1 is x 1. The zeros of f are x 1 and x ±3i.
50. gx x 3 7x 2 x 87
Since 5 2i is a zero, so is 5 2i.
5 2i 1
1
5 2i 1
1
7
5 2i
2 2i
2 2i
5 2i
3
1
14 6i
15 6i
15 6i
15 6i
Section 2.5 Zeros of Polynomial Functions 193
Since are zeros of fx, x 5ix 5i x is a
factor of fx. By long division we have:
2 x ±5i
25
02x 3
x 2 0x 25 ) 2x 3 3x 2 50x 75
2x 3 0x 2 50x
2x 3 3x 2 50x 75
3x 2 50x 75
3x 2 50x 70
Thus, and the zeros of f are and x 3
x ±5i
fx x2 252x 3
Alternate Solution
Since are zeros of fx, x 2ix 2i x is a factor
of fx. By long division we have:
2 x ±2i
4
x 2 0x 4 ) 2x 4 x 3 7x 2 4x 4
Thus,
The zero of x 3 is x 3. The zeros of f are x 3, 5 ± 2i.
0
87
87
0
2x 4 0x 3 8x 2
2 x 2 x 1
x 3 x 2 4x
x 3 0x 2 4x
x 2 0x 4
x 2 0x 4
fx x 2 42x 2 x 1
fx x 2ix 2i2x 1x 1
and the zeros of are x ±2i, x 1
, and x 1.
fx
0
2
2 .