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

111.
9,000,000 0.0001x 2 60x 150,000
Thus, 0 0.0001x 2 60x 9,150,000.
x
P R C xp C
x140 0.0001x 80x 150,000
0.0001x 2 60x 150,000
60 ± 60
0.0002
300,000 ± 10,00015i
Since the solutions are both complex, it is not possible
to determine a price p that would yield a profit of
9 million dollars.
113. False. The most nonreal complex zeros it can have is
two and the Linear Factorization Theorem guarantees
that there are 3 linear factors, so one zero must be real.
Section 2.5 Zeros of Polynomial Functions 203
112. (a)
A 0.0167t 3 0.508t 2 5.60t 13.4
(b) 12
The model is a good fit to
the actual data.
7
0
(c) A 8.5 when t 10 which corresponds to the
year 2000.
(d) A 9 when t 11 which corresponds to the
year 2001.
13
(e) Yes. The degree of A is odd and the leading
coefficient is positive, so as x increases, A will increase.
This implies that attendance will continue to grow.
114. False. f does not have real coefficients.
115. gx fx. This function would have the same zeros 116. gx 3f x. This function has the same zeros as f
as fx so r1, r2, and r3 are also zeros of gx.
because it is a vertical stretch of f. The zeros of g are
r1, r2, and r3. 117. gx fx 5. The graph of gx is a horizontal shift 118. Note that x is a zero of g if and only if
of the graph of fx five units to the right so the zeros
is a zero of f. The zeros of g are and
of gx are 5 r1 , 5 r2 , and 5 r3 .
r3 2 .
r1 2 , r2 2 ,
gx f 2x.
2x
119. gx 3 fx. Since gx is a vertical shift of the graph 120. gx f x. Note that x is a zero of g if and only if
of fx, the zeros of gx cannot be determined.
x is a zero of f. The zeros of g are r1, r2, and r3. 121.
fx x 4 4x 2 k
x 2 4 ± 42 41k
21
x ±2 ± 4 k
4 ± 24 k
2
2 ± 4 k
(a) For there to be four distinct real roots, both and must be positive. This occurs
when Thus, some possible k-values are k etc.
1
4 k 2 ± 4 k
0 < k < 4.
k 1, k 2, k 3, , k 2,
(b) For there to be two real roots, each of multiplicity 2, 4 k must equal zero. Thus, k 4.
(c) For there to be two real zeros and two complex zeros, must be positive and
must be negative. This occurs when Thus, some possible k-values are k 1, k 2, k etc.
1
2 4 k
2 4 k
k < 0.
(d) For there to be four complex zeros, 2 ± 4 k must be nonreal. This occurs when k > 4. Some possible
k-values are k 5, k 6, k 7.4, etc.
122. (a) gx f x 2 (b) gx f 2x
No. This function is a horizontal shift of f x.
No. Since x is a zero of g if and only if 2x is a zero of f,
Note that x is a zero of g if and only if x 2 is a
the number of real and complex zeros of g is the same as
zero of f; the number of real and complex zeros is
not affected by a horizontal shift.
the number of real and complex zeros of f.
2
2 ,