College Algebra 9th txtbk

298 CHAPTER 4 POLYNOMIAL AND RATIONAL FUNCTIONS
88. The solutions to the equation x 3 8 0 are all the cube roots
of 8.
(A) 2 is obviously a cube root of 8; find all others.
(B) How many distinct cube roots of 8 are there?
89. Give a reason for each step in the proof of the rational zero
theorem, assuming that P(x) has degree two.
Step 1: a 2 ( b c) 2 a 1 ( b c) a 0 0
Step 2: a 2 b 2 a 1 bc a 0 c 2 0
Step 3: a 2 b 2 a 1 bc a 0 c 2
Step 4: b is a factor of a 0 c 2 , so b is a factor of a 0 .
Step 5: Modify steps 3 and 4 to conclude that c is a factor of a 2 .
90. Explain how the ideas in Problem 89 can be adapted to give a
proof of the rational zero theorem for P(x) of degree n.
91. Given P(x) x 2 2ix 5 with 2 i a zero, show that 2 i is
not a zero of P(x). Does this contradict Theorem 3? Explain.
92. If P(x) and Q(x) are two polynomials of degree n, and if
P(x) Q(x) for more than n values of x, then how are P(x) and Q(x)
related? [Hint: Consider the polynomial D(x) P(x) Q(x) .]
APPLICATIONS
Find all rational solutions exactly, and find irrational solutions
to one decimal place.
93. STORAGE A rectangular storage unit has dimensions 1 by 2 by
3 feet. If each dimension is increased by the same amount, how
much should this amount be to create a new storage unit with volume
10 times the old?
94. CONSTRUCTION A rectangular box has dimensions 1 by 1 by
2 feet. If each dimension is increased by the same amount, how
much should this amount be to create a new box with volume six
times the old?
95. PACKAGING An open box is to be made from a rectangular
piece of cardboard that measures 8 by 5 inches, by cutting out
squares of the same size from each corner and bending up the sides
(see the figure). If the volume of the box is to be 14 cubic inches,
how large a square should be cut from each corner? [Hint: Determine
the domain of x from physical considerations before starting.]
96. FABRICATION An open metal chemical tank is to be made from
a rectangular piece of stainless steel that measures 10 by 8 feet, by
cutting out squares of the same size from each corner and bending
up the sides (see the figure for Problem 95). If the volume of the
tank is to be 48 cubic feet, how large a square should be cut from
each corner?
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4-4 Rational Functions and Inequalities
Z Rational Functions and Properties of Their Graphs
Z Vertical and Horizontal Asymptotes
Z Analyzing the Graph of a Rational Function
Z Rational Inequalities
In Section 4-4, we will apply our knowledge of graphs and zeros of polynomial functions to
study the graphs of rational functions, that is, functions that are quotients of polynomials.
Our goal will be to produce hand sketches that clearly show all of the important features
of the graph.
Z Rational Functions and Properties of Their Graphs
The number
function
7
13
is called a rational number because it is a quotient (or ratio) of integers. The
f (x) x 1
x 2 x 6
is called a rational function because it is a quotient of polynomials.