College Algebra 9th txtbk

APPENDIX A A-7
In Problems 53 and 54, factor each polynomial in two ways:
(A) As a product of linear factors (with real coefficients) and
quadratic factors (with real coefficients and imaginary zeros).
(B) As a product of linear factors with complex coefficients.
53. P(x) x 4 9x 2 18
54. P(x) x 4 23x 2 50
55. Graph f and indicate any horizontal, vertical, or oblique
asymptotes with dashed lines:
56. Let P(x) x 4 28x 3 262x 2 922x 1.083. Approximate
(to two decimal places) the x intercepts and the local
extrema.
57. Find a polynomial of lowest degree with leading coefficient 1
that has zeros 1 (multiplicity 2), 0 (multiplicity 3), 3 5i,
and 3 5i. Leave the answer in factored form. What is the degree
of the polynomial?
58. If P(x) is a fourth-degree polynomial with integer coefficients
and if i is a zero of P(x), can P(x) have any irrational zeros?
Explain.
59. Let P(x) x 4 9x 3 500x 2 20,000.
(A) Find the smallest positive integer multiple of 10 and the
largest negative integer multiple of 10 that, by Theorem 1
in Section 4-2, are upper and lower bounds, respectively,
for the real zeros of P(x).
(B) Approximate the real zeros of P(x) to two decimal places.
60. Find all zeros (rational, irrational, and imaginary) exactly for
P(x) x 5 4x 4 3x 3 10x 2 10x 12
and factor P(x) into linear factors.
61. Find rational roots exactly and irrational roots to two decimal
places for
P(x) x 5 4x 4 x 3 11x 2 8x 4
62. Give an example of a rational function f(x) that satisfies the following
conditions: the real zeros of f are 5 and 8; x 1 is the
only vertical asymptote; and the line y 3 is a horizontal
asymptote.
63. Use natural logarithms to solve for n.
64. Solve ln y 5x ln A for y. Express the answer in a form that
is free of logarithms.
65. Solve for x.
66. Solve x3 x
x 3 8 0.
f (x) x2 4x 8
x 2
(1 i) n 1
A P
i
y ex 2e x
2
67. Solve (to three decimal places)
APPLICATIONS
68. SHIPPING A mailing service provides customers with
rectangular shipping containers. The length plus the girth of one of
these containers is 10 feet (see the figure). If the end of the container
is square and the volume is 8 cubic feet, find the side length
of the end. Find solutions exactly; round irrational solutions to two
decimal places.
x
69. GEOMETRY The diagonal of a rectangle is 2 feet longer than
one of the sides, and the area of the rectangle is 6 square feet. Find
the dimensions of the rectangle to two decimal places.
70. POPULATION GROWTH If the Democratic Republic of the
Congo has a population of about 60 million people and a doubling
time of 23 years, find the population in
(A) 5 years (B) 30 years
Compute answers to three significant digits.
x
4x
x 2 1 6 3
Length
Girth
71. COMPOUND INTEREST How long will it take money invested
in an account earning 7% compounded annually to double? Use the
annual compounding growth model P P 0 (1 r) t , and compute
the answer to three significant digits.
72. COMPOUND INTEREST Repeat Problem 71 using the continuous
compound interest model P P 0 e rt .
73. EARTHQUAKES If the 1906 and 1989 San Francisco earthquakes
registered 8.3 and 7.1, respectively, on the Richter scale,
how many times more powerful was the 1906 earthquake than
the 1989 earthquake? Use the formula M 2 where
E 0 10 4.40 3 log (EE 0 ),
joules, and compute the answer to one decimal
place.
74. SOUND If the decibel level at a rock concert is 88, find the intensity
of the sound at the concert. Use the formula D 10 log (II 0 ),
where I 0 10 12 watts per square meter, and compute the answer
to two significant digits.
75. ASTRONOMY The square of the time t required for a planet to
make one orbit around the sun varies directly as the cube of its
mean (average) distance d from the sun. Write the equation of variation,
using k as the constant of variation.
y