Algebra

Contents......Page 3
3 Exchange of terms in multiplication ......Page 6
4 Addition in the decimal number system ......Page 7
5 The multiplication table and the multiplication algorithm ......Page 10
6 The division algorithm ......Page 11
7 The binary system ......Page 13
9 The associative law ......Page 16
10 The use of parentheses ......Page 18
11 The distributive law ......Page 19
12 Letters in algebra ......Page 20
13 The addition of negative numbers ......Page 22
14 The multiplication of negative numbers ......Page 23
15 Dealing with fractions ......Page 26
16 Powers ......Page 30
17 Big numbers around us ......Page 31
18 Negative powers ......Page 32
19 Small numbers around us ......Page 34
20 How to multiply am by an. orwhy our definition is convenient ......Page 35
21 The rule of multiplication for powers ......Page 37
22 Formula for short multiplication: The square of a sum ......Page 38
23 How to explain the square of the sum formula to your younger brother or sister ......Page 39
24 The difference of squares ......Page 41
25 The cube of the sum formula ......Page 44
26 The formula for (a + 6)4 ......Page 45
27 Formulas for (a + 6)5, (a + 6)6, . . . and Pascal's triangle ......Page 47
28 Polynomials ......Page 49
29 A digression: When are polynomials equal? ......Page 51
30 How many monomials do we get? ......Page 53
31 Coefficients and values ......Page 54
32 Factoring ......Page 56
34 Converting a rational expression intothe quotient of two polynomials ......Page 61
35 Polynomial and rational fractions in one variable ......Page 66
36 Division of polynomials in one variable: the remainder ......Page 67
37 The remainder when dividing by % - a ......Page 73
38 Values of polynomials, and interpolation ......Page 77
39 Arithmetic progressions ......Page 82
40 The sum of an arithmetic progression ......Page 84
41 Geometric progressions ......Page 86
42 The sum of a geometric progression ......Page 88
43 Different problems about progressions ......Page 90
44 The well-tempered clavier ......Page 92
45 The sum of an infinite geometric progression ......Page 96
46 Equations ......Page 99
48 Quadratic equations ......Page 100
49 The case p = O . Square roots ......Page 101
50 Rules for square roots ......Page 104
51 The equation x^2 + px + q = 0 ......Page 105
52 Vieta's theorem ......Page 107
53 Factoring ax^2 + bx + c ......Page 111
54 A formula for ax^2 + bx + c = 0 ......Page 112
56 A quadratic equation becomes linear ......Page 113
57 The graph of the quadratic polynomial ......Page 115
59 Maximum and minimum values of aquadratic polynomial ......Page 119
60 Biquadratic equations ......Page 121
61 Symmetric equations ......Page 122
62 How to confuse students on an exam ......Page 123
63 Roots ......Page 125
64 Non-integer powers ......Page 128
65 Proving inequalities ......Page 132
66 Arithmetic and geometric means ......Page 135
68 Problems about maximum and minimum ......Page 137
69 Geometric illustrations ......Page 139
70 The arithmetic and geometric meansof several numbers ......Page 141
71 The quadratic mean ......Page 149
72 The harmonic mean ......Page 152