## AlgebraThis book is about algebra. This is a very old science and its gems have lost their charm for us through everyday use. We have tried in this book to refresh them for you. The main part of the book is made up of problems. The best way to deal with them is: Solve the problem by yourself - compare your solution with the solution in the book (if it exists) - go to the next problem. However, if you have difficulties solving a problem (and some of them are quite difficult), you may read the hint or start to read the solution. If there is no solution in the book for some problem, you may skip it (it is not heavily used in the sequel) and return to it later. The book is divided into sections devoted to different topics. Some of them are very short, others are rather long. Of course, you know arithmetic pretty well. However, we shall go through it once more, starting with easy things. 2 Exchange of terms in addition Let's add 3 and 5: 3+5=8. And now change the order: 5+3=8. We get the same result. Adding three apples to five apples is the same as adding five apples to three - apples do not disappear and we get eight of them in both cases. 3 Exchange of terms in multiplication Multiplication has a similar property. But let us first agree on notation. |

### What people are saying - Write a review

I am using this book as the guiding text for teaching a teen who missed much of his mathematical instruction prior to high school. It is ideal because the authors carefully build the necessary mathematical understandings from one problem to the next.

This text will be completed in two years with this boy. I decided to leave chapters 33 through 45 for his second year of algebra, skipping ahead as we approach the end of the first year (Algebra 1). We picked up with chapter 46 "Equations" with a plan to do as much as we can fit in before pausing this summer.

I have not skipped a single problem. I have been tempted, but every time I have been glad that I persevered. Every problem has something to teach.

I love the absence of distracting photographs, sidebars, and such. If I feel that something of nature will add to the instruction, I find my own and add it into the assignment. By the time I finish with this boy, I will have a wonderful set of assignments all recorded and ready to go for another student. There are being developed and stored on WPIs software, ASSISTments.

This book is about algebra. This is a very old science and its gems have lost their charm for us through everyday use. We have tried in this book to refresh them for you. The main part of the book is made up of problems. The best way to deal with them is: Solve the problem by yourself - compare your solution with the solution in the book (if it exists) - go to the next problem. However, if you have difficulties solving a problem (and some of them are quite difficult), you may read the hint or start to read the solution. If there is no solution in the book for some problem, you may skip it (it is not heavily used in the sequel) and return to it later. The book is divided into sections devoted to different topics. Some of them are very short, others are rather long. Of course, you know arithmetic pretty well. However, we shall go through it once more, starting with easy things. 2 Exchange of terms in addition Let's add 3 and 5: 3+5=8. And now change the order: 5+3=8. We get the same result. Adding three apples to five apples is the same as adding five apples to three - apples do not disappear and we get eight of them in both cases. 3 Exchange of terms in multiplication Multiplication has a similar property. But let us first agree on notation.

### Contents

1 Introduction | 1 |

4 Addition in the decimal number system | 2 |

5 The multiplication table and the multiplication algorithm | 5 |

6 The division algorithm | 6 |

7 The binary system | 8 |

8 The commutative law | 11 |

10 The use of parentheses | 13 |

11 The distributive law | 14 |

38 Values of polynomials and interpolation | 72 |

39 Arithmetic progressions | 77 |

40 The sum of an arithmetic progression | 79 |

41 Geometric progressions 41 Geometric progressions | 81 |

42 The sum of a geometric progression | 83 |

43 Different problems about progressions | 85 |

44 The welltempered clavier | 87 |

45 The sum of an infinite geometric progression | 91 |

12 Letters in algebra | 15 |

13 The addition of negative numbers | 17 |

14 The multiplication of negative numbers | 18 |

15 Dealing with fractions | 21 |

16 Powers | 25 |

17 Big numbers around us | 26 |

18 Negative powers | 27 |

19 Small numbers around us | 29 |

20 How to multiply am by an or why our definition is convenient | 30 |

21 The rule of multiplication for powers | 32 |

The square of a sum | 33 |

23 How to explain the square of the sum formula to your younger brother or sister | 34 |

24 The difference of squares | 36 |

25 The cube of the sum formula | 39 |

26 The formula for a + 64 | 40 |

27 Formulas for a + 65 a + 66 and Pascals triangle | 42 |

28 Polynomials | 44 |

When are polynomials equal? | 46 |

30 How many monomials do we get? | 48 |

31 Coefficients and values | 49 |

32 Factoring | 51 |

33 Rational expressions | 56 |

35 Polynomial and rational fractions in one variable | 61 |

36 Division of polynomials in one variable the remainder | 62 |

37 The remainder when dividing by x a | 68 |

46 Equations | 94 |

47 A short glossary | 95 |

49 The case p 0 Square roots | 96 |

50 Rules for square roots | 99 |

51 The equation x2 + px + q 0 Problem 235 Solve the equation | 100 |

52 Vietas theorem | 102 |

53 Factoring ax2 f bx + c | 106 |

54 A formula for ax2 + bx + c 0 where a 0 | 107 |

55 One more formula concerning quadratic equations | 108 |

57 The graph of the quadratic polynomial | 110 |

58 Quadratic inequalities | 114 |

60 Biquadratic equations | 116 |

61 Symmetric equations | 117 |

62 How to confuse students on an exam | 118 |

63 Roots | 120 |

64 Noninteger powers | 123 |

65 Proving inequalities | 127 |

66 Arithmetic and geometric means | 130 |

67 The geometric mean does not exceed the arithmetic mean | 132 |

69 Geometric illustrations | 134 |

70 The arithmetic and geometric means of several numbers | 136 |

71 The quadratic mean | 144 |

72 The harmonic mean | 147 |