SCHAUM’S Easy OUTLINES

129 Pages·2007·1.57 MB·English

SCHAUM’S Easy OUTLINES
P
RECALCULUS
Other Books in Schaum’s
Easy Outlines Series Include:
Schaum’s Easy Outlines:College Mathematics
Schaum’s Easy Outlines:College Algebra
Schaum’s Easy Outlines:Calculus
Schaum’s Easy Outlines:Elementary Algebra
Schaum’s Easy Outlines:Mathematical Handbook
of Formulas and Tables
Schaum’s Easy Outlines:Geometry
Schaum’s Easy Outlines:Trigonometry
Schaum’s Easy Outlines:Probability and Statistics
Schaum’s Easy Outlines:Statistics
Schaum’s Easy Outlines:Principles of Accounting
Schaum’s Easy Outlines:Biology
Schaum’s Easy Outlines:College Chemistry
Schaum’s Easy Outlines:Genetics
Schaum’s Easy Outlines:Human Anatomy
and Physiology
Schaum’s Easy Outlines:Organic Chemistry
Schaum’s Easy Outlines:Physics
Schaum’s Easy Outlines:Basic Electricity
Schaum’s Easy Outlines:Programming with C++
Schaum’s Easy Outlines:Programming with Java
Schaum’s Easy Outlines:French
Schaum’s Easy Outlines:German
Schaum’s Easy Outlines:Spanish
Schaum’s Easy Outlines:Writing and Grammar
SCHAUM’S Easy OUTLINES
P
RECALCULUS
Based on Schaum’s Outline of
Precalculus
by Fred Safier
Abridgement Editor:
Kimberly S. Kirkpatrick
SCHAUM’S OUTLINE SERIES
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DOI: 10.1036/0071398406
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Contents
Chapter 1 Number Systems, Polynomials,
and Exponents 1
Chapter 2 Equations and Inequalities 13
Chapter 3 Systems of Equations and
Partial Fractions 27
Chapter 4 Analytic Geometry and Functions 39
Chapter 5 Algebraic Functions and
Their Graphs 54
Chapter 6 Exponential and
Logarithmic Functions 71
Chapter 7 Conic Sections 78
Chapter 8 Trigonometric Functions 87
Chapter 9 Trigonometric Identities and
Trigonometric Inverses 104
Chapter 10 Sequences and Series 116
Index 121
v
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Chapter 1
Number Systems,
Polynomials,
and Exponents
In This Chapter:
✔ Sets of Numbers
✔ Axioms for the Real Number System
✔ Properties of Inequalities
✔ Absolute Value
✔ Complex Numbers
✔ Order of Operations
✔ Polynomials
✔ Factoring
✔ Exponents
✔ Rational and Radical Expressions
Sets of Numbers
The sets of numbers used in algebra are,in general,subsets of R,the set
of real numbers.
• Natural numbersN:The counting numbers,e.g.,1,2,3,…
• IntegersZ:The counting numbers,together with their opposites
and 0,e.g.,0,1,2,3,…,−1,−2,−3…
1
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2 PRECALCULUS
• Rational NumbersQ:The set of all numbers that can be written
as quotients a/b,b ≠0,aand bintegers,e.g.,3/17,−5/13,…
• Irrational Numbers H:All real numbers that are not rational
numbers,e.g.,π, 2,35, −π/3,…
Example 1.1: The number −5 is a member of the
sets Z,Q,R. The number 156.73 is a member of the
sets Q, R. The number 5π is a member of the sets
H,R.
Axioms for the Real Number System
There are two fundamental operations,addition and multiplication,which
have the following properties (a,b,carbitrary real numbers):
• Commutative Laws :
a +b =b +a:order does not matter in addition.
ab =ba:order does not matter in multiplication.
• Associative Laws:
a + (b + c) = (a + b) + c: grouping does not matter in
repeated addition.
a(bc) =(ab)c:grouping does not matter in repeated multi-
plication.
• Distributive Laws:
a(b +c) =ab +ac; also (a +b)c =ac +bc:multiplication is
distributive over addition.
• Zero Factor Laws:
For every real number a,a ⋅ 0 =0.
If ab =0,then either a =0 or b =0.
• Laws for Negatives:
−(−a) =a
(−a)(−b) =ab
−ab =(−a)b =a(−b) =−(−a)(−b) =−(ab)
(−1)a =−a
• Laws for Quotients:
a −a a −a
− = = =−
b b −b −b
−a a
=
−b b
a c
= if and only if ad=bc.
b d
CHAPTER 1:Number Systems, Polynomials, and Exponents 3
a ka
= ,for k any nonzero real number.
b kb
Properties of Inequalities
The number a is less than b,written a <b,if b – ais positive. If a <bthen
b is greater than a,written b >a. If ais either less than or equal to b,this
is written a ≤ b. If a ≤ b then bis greater than or equal to a,written b ≥ a.
The following properties may be deduced from these definitions:
• If a <b,then a +c <b +c.
ac<bc if c>0
• If a <b,then 
ac>bc if c<0.
• If a <band b <c,then a <c.
Absolute Value
The absolute valueof a real number a,written (cid:1)a(cid:1),is defined as follows:
a if a≥0
a = .
−a if a<0.
Complex Numbers
Not all numbers are real numbers. The set of complex numbers,C,con-
tains all numbers of the standard form a +bi,where aand bare real and
i2=−1. Since every real number xcan be written as x +0i,it follows that
every real number is also a complex number. The complex numbers that
are notreal are sometimes known as imaginary numbers.
1 3
Example 1.2: 3+ −4 =3+2i, −5i, 2πi, + i are examples of
2 2
complex (imaginary) numbers.
In standard form,complex numbers can be combined using the op-
erations defined for real numbers, together with the definition of the
imaginary unit i:i2=−1. The conjugate of a complex number zis denot-
ed ¯z. If z =a+bi,then ¯z =a−bi.