Advanced Algebra

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Ginn, 1905 - 285 páginas
This book is designed for use in secondary schools and in short college courses. It aims to present in concise but clear form the portions of algebra that are required for entrance to the most exacting colleges and technical schools. The chapters in 'Algebra to Quadratics' are intended for a review of the subject. The rest of the text concentrates on subjects that are most vital, which is why topics that demand a knowledge of calculus for complete comprehension have been omitted.

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Contenido

Multiplication
9
Multiplication of Monomials by Polynomials
10
Types of Multiplication
11
The Square of a Polynomial
12
Division of Monomials
13
Types of Division
15
FACTORING SECTION PAGE 28 Statement of the Problem
16
Factoring by grouping Terms
17
Factors of a Quadratic Trinomial
18
Factoring the Difference of Squares
20
Replacing a Parenthesis by a Letter
21
Factoring Binomials of the Form an + bn
22
H C F of Two Polynomials
23
Method of finding the H C F of Two Polynomials
24
Least Common Multiple
26
CHAPTER III
27
Least Common Denominators of Several Fractions
28
Addition of Fractions
29
CHAPTER IV
32
Linear Equations in One Variable
33
Solution of Problems
37
Linear Equations in Two Variables
40
SECTION PAGE 58 Independent Equations
41
Solution of a Pair of Simultaneous Linear Equations
42
Résumé
43
Solution of Problems involving Two Unknowns
45
Solution of Linear Equations in Several Variables
47
CHAPTER V
49
Theorem
50
CHAPTER VI
52
The Practical Necessity for Irrational Numbers
53
Extraction of Square Root of Numbers
54
Approximation of Irrational Numbers
55
Sequences
56
Notation
57
Reduction of a Radical to its Simplest Form
58
Addition and Subtraction of Radicals
59
Multiplication and Division of Radicals
60
Rationalization
61
Solution of Equations involving Radicals
63
CHAPTER VII
66
Further Assumptions
67
Operations with Radical Polynomials
69
QUADRATICS AND BEYOND CHAPTER VIII
70
Pure Quadratics
72
Solution of Quadratic Equations by Factoring
75
Quadratic Form
77
Problems solvable by Quadratic Equations
79
Theorems regarding Quadratic Equations
82
Theorem
83
Theorem
84
CHAPTER IX
87
Cartesian Coördinates
88
The Graph of an Equation
90
Restriction to Coördinates
91
Plotting Equations after Solution
93
Graph of the Linear Equation
94
Method of plotting a Line from its Equation
96
Solution of Linear Equations and the Intersection of their Graphs
97
Graphs of Dependent Equations
99
Graph of the Quadratic Equation
100
Form of the Graph of a Quadratic Equation
101
The Special Quadratic ax2 + bx 0
103
The Special Quadratic ax2 + c 0
104
Sum and Difference of Roots
106
Variation in Sign of a Quadratic
107
CHAPTER X
111
Number of Solutions
113
Solution when neither Equation is Linear
114
Equivalence of Pairs of Equations
120
Incompatible Equations
121
Graphical Representation of Simultaneous Quadratic Equations
122
Graphical Meaning of Homogeneous Equations
123
CHAPTER XI
125
Permutations
144
Combinations
146
Circular Permutations
149
Theorem
150
CHAPTER XVI
152
Addition and Subtraction of Imaginary Numbers
153
Multiplication and Division of Imaginaries
154
Complex Numbers
155
Addition and Subtraction
156
Multiplication of Complex Numbers
157
Conjugate Complex Numbers
158
Polar Representation
160
Powers of Numbers in Polar Form
161
Division in Polar Form
162
CHAPTER XVII
166
Synthetic Division
167
Proof of the Rule for Synthetic Division
169
Plotting of Equations
170
Extent of the Table of Values
171
SECTION PAGE 165 Roots of an Equation
172
Graphical Interpretation
174
Graphical Interpretation of Imaginary Roots
175
Relation between Roots and Coefficients
177
The General Term in the Binomial Expansion
178
Properties of Binomial Surds
179
Formation of Equations
180
To multiply the Roots by a Constant
183
Descartes Rule of Signs
186
Negative Roots
189
Integral Roots
190
Diminishing the Roots of an Equation
191
Graphical Interpretation of Decreasing Roots
193
Location Principle
194
Approximate Calculation of Roots by Horners Method
195
Roots nearly Equal
200
CHAPTER XVIII
203
Solution of Three Linear Equations
204
Inversion
208
Number of Terms
210
Multiplication by a Constant
213
Interchange of Rows or Columns
214
Identical Rows or Columns
215
Sum of Determinants
216
Vanishing of a Determinant
217
Evaluation by Factoring
218
Practical Directions for evaluating Determinants
219
Solution of Linear Equations
221
Solution of Homogeneous Linear Equations
223
CHAPTER XIX
225
Development when x 0 has Imaginary Roots
229
Development when Qx x a
232
General Case
233
CHAPTER XX
235
Logarithms
236
Operations on Logarithms
237
Common System of Logarithms
239
Use of Tables
241
Interpolation
242
Antilogarithms
243
Cologarithms
245
Change of Base
247
Exponential Equations
251
Compound Interest
253
CHAPTER XXI
256
Convergents
258
Recurring Continued Fractions
260
Expression of a Surd as a Recurring Continued Fraction
263
Properties of Convergents
265
Limit of Error
267
CHAPTER XXII
269
Conditional Linear Inequalities
271
VARIATION
273
CHAPTER XXV
279
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Página 235 - The characteristic of the logarithm of a number greater than 1 is a positive integer or zero, and is one less than the number of digits to the left of the decimal point.
Página 51 - Find the square root of the first term, write it as the first term of the root, and subtract its square from the given polynomial. Divide the first term of the remainder by...
Página 85 - Pythagorean theorem, which states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides.
Página 52 - Separate the given number into periods of two figures each, beginning from the units' place. 2d. find the greatest number whose square is contained in the left-hand period ; this is the first figure of the required root. >Subtract its square from the first period, and to the remainder bring down the second period for a dividend.
Página 13 - Multiply the divisor by the first term of the quotient and subtract the product from the dividend.
Página 155 - The number of permutations of n things taken r at a time is n!
Página 12 - The square of any polynomial equals the sum of the squares of the terms plus twice the product of each term by each term which follows it.
Página 51 - Divide the first term of the remainder by twice the first term of the root, and add the quotient to the part of the root already found, and also to the trial-divisor.
Página 51 - We next multiply the complete divisor by the last term of the root and subtract the product from the last remainder.
Página 25 - The numerator and denominator of a fraction may be multiplied or divided by the same number without changing the value of the fraction.

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