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SCHAUMS Easy OUTLINES

PRECALCULUS

Other Books in SchaumsEasy Outlines Series Include:

Schaums Easy Outlines: College MathematicsSchaums Easy Outlines: College AlgebraSchaums Easy Outlines: CalculusSchaums Easy Outlines: Elementary AlgebraSchaums Easy Outlines: Mathematical Handbook

of Formulas and TablesSchaums Easy Outlines: GeometrySchaums Easy Outlines: TrigonometrySchaums Easy Outlines: Probability and StatisticsSchaums Easy Outlines: StatisticsSchaums Easy Outlines: Principles of AccountingSchaums Easy Outlines: BiologySchaums Easy Outlines: College ChemistrySchaums Easy Outlines: GeneticsSchaums Easy Outlines: Human Anatomy

and PhysiologySchaums Easy Outlines: Organic ChemistrySchaums Easy Outlines: PhysicsSchaums Easy Outlines: Basic ElectricitySchaums Easy Outlines: Programming with C++Schaums Easy Outlines: Programming with JavaSchaums Easy Outlines: FrenchSchaums Easy Outlines: German Schaums Easy Outlines: SpanishSchaums Easy Outlines: Writing and Grammar

SCHAUMS Easy OUTLINES

PRECALCULUS

B a s e d o n S c h a u m s Out l ine o f

Precalcu lus

b y F r e d S a f i e r

A b r i d g e m e n t E d i t o r :

K i m b e r ly S . K i r k p a t r i c k

S C H AU M S O U T L I N E S E R I E SM c G R AW - H I L L

New York Chicago San Francisco Lisbon London Madrid

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Seoul Singapore Sydney Toronto

Copyright 2002 by The McGraw-Hill Companies, Inc. All rights reserved. Manufactured in theUnited States of America. Except as permitted under the United States Copyright Act of 1976, no partof this publication may be reproduced or distributed in any form or by any means, or stored in a data-base or retrieval system, without the prior written permission of the publisher.

0-07-139840-6

The material in this eBook also appears in the print version of this title: 0-07-138340-9

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DOI: 10.1036/0071398406

Contents

Chapter 1 Number Systems, Polynomials, and Exponents 1

Chapter 2 Equations and Inequalities 13Chapter 3 Systems of Equations and

Partial Fractions 27Chapter 4 Analytic Geometry and Functions 39Chapter 5 Algebraic Functions and

Their Graphs 54Chapter 6 Exponential and

Logarithmic Functions 71Chapter 7 Conic Sections 78Chapter 8 Trigonometric Functions 87Chapter 9 Trigonometric Identities and

Trigonometric Inverses 104Chapter 10 Sequences and Series 116Index 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 setof real numbers.

Natural numbers N: The counting numbers, e.g., 1, 2, 3, Integers Z: The counting numbers, together with their opposites

and 0, e.g., 0, 1, 2, 3, , 1, 2, 3

1Copyright 2002 by the Mcgraw-Hill Companies, Inc. Click Here for Terms of Use.

Rational Numbers Q: The set of all numbers that can be writtenas quotients a/b, b 0, a and b integers, e.g., 3/17, 5/13,

Irrational Numbers H: All real numbers that are not rationalnumbers, e.g., , /3,

Example 1.1: The number 5 is a member of thesets Z, Q, R. The number 156.73 is a member of thesets Q, R. The number 5 is a member of the sets H, R.

Axioms for the Real Number System

There are two fundamental operations, addition and multiplication, whichhave the following properties (a, b, c arbitrary 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 isdistributive 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

b

a

b

a

b

a

ba

b

a

ba

b

c

dad bcif and only if .

2 53, ,

2 PRECALCULUS

Properties of Inequalities

The number a is less than b, written a < b, if b a is positive. If a < b thenb is greater than a, written b > a. If a is either less than or equal to b, thisis written a b. If a b then b is 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.

If a < b, then.

If a < b and b < c, then a < c.

Absolute Value

The absolute value of a real number a, written a, is defined as follows:

.

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 a and b are real andi2 = 1. Since every real number x can be written as x + 0i, it follows thatevery real number is also a complex number. The complex numbers thatare not real are sometimes known as imaginary numbers.

Example 1.2: are examples of

complex (imaginary) numbers.

In standard form, complex numbers can be combined using the op-erations defined for real numbers, together with the definition of theimaginary unit i: i2 = 1. The conjugate of a complex number z is denot-ed z. If z = a + bi, then z = a bi.

3 4 3 2 5 21

2

3

2+ = + +i i i i, , ,

aa a

a a=

> 0, x1/n is the positive real number y which, when

raised to the nth power gives x.

3 2 3 31

21

3

1 3 2

32 4 4 5 2

24

4 52

4

2

2

4 5x y x y z xy

x yz

y

x

z

x y + = + = +( )

( ) ( )

xx

nn

= 1

a b a b a b

a b

a ab b a b

a ab b a b

2 2

2 2

2 2 2

2 2 2

2

2

= + ++ + = + + =

( )( )

( )

( )

Difference of two squares

is prime. Sum of two squares

Square of a sum

Square of a difference

8 PRECALCULUS

if x = 0, x1/n = 0. if x < 0, x1/n is not a real number.

Remember

Even roots of negative numbersare not real numbers.

Example 1.18: (a) 161/4 = 2; (b) 161/4 = (16)1/4 = 2; (c) ( 16)1/4is not a real number; (d ) ( 8)1/3 = 2

xm/n is defined by: xm/n = (x1/n)m = (xm)1/n, provided x1/n is real.

Example 1.19: (a) , (b) ( 64)5/6 is nota real number.

Laws of exponents for a and b rational nu

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