Ә.Н.Шыныбеков. Алгебра 8-сынып, 2012

Foreword

Review of Grade 7 material

Chapter I. Square root

1. Definition of a square root

1.1. Concept of a square root

1.2. Arithmetic square root

2. Concept of an irrational number

2.1. Approximate calculation of a square root

2.2. Irrational numbers. Set of real numbers

3. Correspondence between real numbers and points on a line

3.1. Integer and fractional parts of a number

3.2. Correspondence between real numbers and points on a line

3.3. Some numerical intervals and their representation by inequalities

4. Properties of square roots

4.1. Properties of square roots

4.2. Transforming expressions containing square roots

5. Graph of the square-root function

5.1. The square-root function and its graph

5.2. Application of square roots in geometry

Additional problems for Chapter I

Chapter II. Quadratic equations

1. Quadratic equation and its roots

1.1. Definition of a quadratic equation

1.2. Solving incomplete quadratic equations

2. Quadratic formula

2.1. General form of the formula

2.2. Formula when b is even

3. Vieta's theorem

3.1. Vieta's theorem

3.2. Converse theorem

3.3. The case when a plus or minus b plus c equals zero

4. Properties of the roots of a quadratic equation

4.1. Studying the roots of quadratic equations

4.2. Factoring a quadratic trinomial

5. Rational equations. Problems reducible to quadratic equations

5.1. Rational equations

5.2. Problems reducible to a quadratic equation

6. Solving systems of second-degree equations

Additional problems for Chapter II

Chapter III. Quadratic function

1. Quadratic function and its graph

1.1. Graph of the function y equals ax squared

1.2. Graph of y equals a times the square of x minus m

1.3. Graph of y equals a times the square of x minus m plus n

2. Transformations of a function graph

2.1. Translation of a function graph

2.2. Graph of a linear fractional function

2.3. Graphs of functions containing absolute value

3. Some properties of functions

3.1. Zeros of a function and the concept of constant sign

3.2. Intervals on which a function keeps a constant sign

3.3. Intervals of increase and decrease. Function extrema

3.4. Even and odd functions

4. Solving an equation and a system of equations in two variables graphically

4.1. Solving an equation graphically

4.2. Solving a system of equations in two variables graphically

Additional problems for Chapter III

Chapter IV. Quadratic inequalities

1. Proving inequalities

1.1. Properties of numerical inequalities

1.2. Methods of proving inequalities

2. Solving quadratic inequalities

2.1. Concept of solving inequalities

2.2. Solving quadratic inequalities in one variable

2.3. Solving quadratic inequalities in one variable graphically

2.4. Solving systems of inequalities in one variable

3. Solving rational inequalities

4. Representation of plane figures by equations, inequalities in two variables and their systems

4.1. Equations in two variables

4.2. Geometric meaning of inequalities in two variables

5. Statistical characteristics

5.1. Population and sample

5.2. Variance and standard deviation

Additional problems for Chapter IV

Supplementary materials for advanced study of mathematics

Chapter I. Real numbers

1. Natural numbers. Divisibility tests

1.1. Natural numbers and their properties

1.2. Divisibility tests

2. Prime and composite numbers

2.1. Prime and composite numbers

2.2. Fundamental theorem of arithmetic

2.3. Canonical factorization of natural numbers

3. Greatest common divisor and least common multiple

3.1. Greatest common divisor and least common multiple of numbers

3.2. Properties of common divisors and common multiples

4. Division of integers with remainder

4.1. Subtraction. Integers

4.2. Division of integers with remainder

4.3. Euclidean algorithm

5. Solving linear indeterminate equations

5.1. Concept of indeterminate equations

5.2. Solving linear indeterminate equations

6. Pigeonhole principle

Additional problems for Chapter I, advanced level

Supplementary materials and problems for Chapter II

7. Division of a polynomial by a polynomial

8. Remainder theorem. Property of integer roots of polynomials with integer coefficients

Equations and systems of equations

1. Algebraic equations and their roots. Equivalent equations

2. Rational equations

3. Biquadratic equations. Solving equations by introducing a new variable

4. Symmetric equations

5. Solving higher-degree equations by factoring

6. Irrational equations

7. Equations containing absolute value

8. Equations with parameters

9. Methods for solving systems of equations

10. Problems reducible to systems of equations

Additional problems for Chapter III, advanced level

Additional problems for Chapter IV, advanced level

Subject index

Answers