Table of Contents
Preface ix
1 Numbers of Algebra 1
Natural Numbers, Whole Numbers, and Integers 1
Rational, Irrational, and Real Numbers 4
Properties of the Real Numbers 8
2 Computation with Real Numbers 14
Comparing Numbers and Absolute Value 14
Addition and Subtraction of Signed Numbers 17
Multiplication and Division of Signed Numbers 26
3 Roots and Radicals 32
Squares, Square Roots, and Perfect Squares 32
Cube Roots and nth Roots 36
Simplifying Radicals 39
4 Exponentiation 44
Exponents 44
Natural Number Exponents 45
Zero and Negative Integer Exponents 48
Unit Fraction and Rational Exponents 53
5 Order of Operations 58
Grouping Symbols 58
PEMDAS 60
6 Algebraic Expressions 64
Algebraic Terminology 64
Evaluating Algebraic Expressions 66
Dealing with Parentheses 70
7 Rules for Exponents 74
Product Rule 74
Quotient Rule 75
Rules for Powers 77
Rules for Exponents Summary 79
8 Adding and Subtracting Polynomials 83
Terms and Monomials 83
Polynomials 85
Like Terms 87
Addition and Subtraction of Monomials 88
Combining Like Terms 89
Addition and Subtraction of Polynomials 90
9 Multiplying Polynomials 94
Multiplying Monomials 94
Multiplying Polynomials by Monomials 96
Multiplying Binomials 98
The FOIL Method 99
Multiplying Polynomials 101
Special Products 103
10 Simplifying Polynomial Expressions 104
Identifying Polynomial Expressions 104
Simplifying Polynomial Expressions 106
11 Dividing Polynomials 110
Dividing a Polynomial by a Monomial 110
Dividing a Polynomial by a Polynomial 113
12 Factoring Polynomials 119
Factoring and Its Objectives 119
Greatest Common Factor 120
GCF with a Negative Coefficient 123
A Quantity as a Common Factor 125
Factoring Four Terms 126
Factoring Quadratic Trinomials 127
Factoring Perfect Trinomial Squares 133
Factoring Binomials 134
Guidelines for Factoring 137
13 Rational Expressions 139
Reducing Algebraic Fractions to Lowest Terms 139
Multiplying Algebraic Fractions 143
Dividing Algebraic Fractions 145
Adding (or Subtracting) Algebraic Fractions, Like Denominators 146
Adding (or Subtracting) Algebraic Fractions, Unlike Denominators 148
Simplifying Complex Fractions 151
14 Solving Linear Equations and Inequalities 154
Solving One-Variable Linear Equations 154
Solving Two-Variable Linear Equations for a Specific Variable 159
Solving One-Variable Linear Inequalities 160
15 Solving Quadratic Equations 163
Solving Quadratic Equations of the Form ax2 + c = 0 163
Solving Quadratic Equations by Factoring 165
Solving Quadratic Equations by Completing the Square 166
Solving Quadratic Equations by Using the Quadratic Formula 167
16 The Cartesian Coordinate Plane 171
Definitions for the Plane 171
Ordered Pairs in the Plane 171
Quadrants of the Plane 174
Finding the Distance Between Two Points in the Plane 176
Finding the Midpoint Between Two Points in the Plane 177
Finding the Slope of a Line Through Two Points in the Plane 178
Slopes of Parallel and Perpendicular Lines 181
17 Graphing Linear Equations 184
Properties of a Line 184
Graphing a Linear Equation That Is in Standard Form 184
Graphing a Linear Equation That Is in Slope-y-Intercept Form 186
18 The Equation of a Line 189
Determining the Equation of a Line Given the Slope and y-Intercept 189
Determining the Equation of a Line Given the Slope and One Point on the Line 190
Determining the Equation of a Line Given Two Distinct Points on the Line 192
19 Basic Function Concepts 195
Representations of a Function 195
Terminology of Functions 197
Some Common Functions 201
20 Systems of Equations 205
Solutions to a System of Equations 205
Solving a System of Equations by Substitution 206
Solving a System of Equations by Elimination 208
Solving a System of Equations by Graphing 210
21 Signal Words and Phrases 213
Common Signal Words and Phrases for Addition 213
Common Signal Words and Phrases for Subtraction 215
Common Signal Words and Phrases for Multiplication 217
Common Signal Words and Phrases for Division 219
Common Signal Words and Phrases for Equality 221
22 Word Problems 225
Algebraic Problem-Solving Process 225
Be Careful with Units 227
Age Relationships 227
Consecutive Numbers 229
Ratios 231
Mixtures 232
Coins 235
Rate-Time-Distance 237
Work 238
Percentage 240
Simple Interest 243
Number Relationships 244
Geometric Shapes 246
Appendix: Measurement Units and Conversions 249
Answer Key 251
Index 283