Ю.Н.Макарычев, Н.Г.Миндюк. Алгебра 7-сынып, 2007

Foreword

1.1. Set

1.1.1. Set. Types of sets. Ways of defining a set

1.1.2. Subset

1.2. Numerical expressions and expressions with variables

1.2.1. Numerical expressions

1.2.2. Expressions with variables

Additional exercises for Section 1.2

1.3. On mathematical language and mathematical models

1.3.1. What is mathematical language?

1.3.2. What is a mathematical model?

Additional exercises for Section 1.3

2.1. Power with a natural exponent

2.1.1. Definition of a power with a natural exponent

2.1.2. Multiplication and division of powers with natural exponents

2.2. Monomials and operations on them

2.2.1. Standard form of a monomial

2.2.2. Multiplication of monomials

2.2.3. Division of a monomial by a monomial

2.2.4. Raising monomials to a power

2.2.5. Identity

Additional exercises for Sections 2.1–2.2

2.3. Power functions with natural exponents

2.3.1. Power function with an even exponent

2.3.2. Power function with an odd exponent

Additional exercises for Section 2.3

3.1. Standard form of a polynomial

3.1.1. Definition of a polynomial. Evaluating a polynomial

3.1.2. Standard form of a polynomial

3.2. Sum, difference and product of polynomials

3.2.1. Addition and subtraction of polynomials

3.2.2. Multiplication of polynomials

3.2.3. Division of a polynomial by a monomial

3.2.4. Multiplication of a polynomial by a polynomial

3.2.5. Raising a polynomial to a power

Additional exercises for Sections 3.1–3.2

3.3. Solving equations

3.3.1. Equation in one variable and its solution

3.3.2. Solving equations reducible to linear equations

3.3.3. Solving problems using equations

Additional exercises for Section 3.3

3.4. Factoring polynomials

3.4.1. Greatest common divisor of monomials

3.4.2. Factoring out the common factor

3.4.3. Grouping method

3.5. Applications of factoring polynomials

3.5.1. Calculations. Proving identities

3.5.2. Solving equations by factoring polynomials

Additional exercises for Sections 3.4–3.5

3.6. Special product formulas

3.6.1. Difference of squares

3.6.2. Factoring a difference of squares

3.6.3. Square of a sum and a difference

3.6.4. Factoring using square-of-sum and square-of-difference formulas

3.6.5. Quadratic trinomial

3.6.6. Square of a sum of several terms

3.6.7. Cube of a sum and a difference

3.6.8. Formulas for factoring the sum and difference of cubes

3.6.9. Factoring sums and differences of cubes

3.6.10. Factoring differences and sums of nth powers

3.6.11. Applying various factoring methods

Additional exercises for Section 3.6

3.7. Division of a polynomial by a polynomial

3.7.1. Exact division of a polynomial by a polynomial

3.7.2. Division of a polynomial by a polynomial with remainder

3.7.3. Remainder theorem. Root of a polynomial

4.1. Divisibility of numbers

4.1.1. Natural numbers and their properties

4.1.2. Integers and operations on them

4.1.3. Concept of divisibility. Properties of divisibility

4.1.4. Division with remainder

4.1.5. Divisibility of sums and differences

4.1.6. Divisibility tests for 2, 4, 5, 7, 8, 11, 13, 25 and 125

4.1.7. Product of two consecutive natural numbers

4.2. Greatest common divisor and least common multiple. Euclidean algorithm

4.2.1. Greatest common divisor

4.2.2. Euclidean algorithm

4.2.3. Least common multiple

4.2.4. Coprime numbers and their properties

4.3. Prime and composite numbers. Fundamental theorem of arithmetic

4.3.1. Prime and composite numbers

4.3.2. Prime factorization of a natural number. Fundamental theorem of arithmetic

4.4. Applying properties of integers to solving indeterminate equations

4.4.1. Concept of an indeterminate equation

4.4.2. Solving a first-degree linear equation in two variables over the integers

4.4.3. Systems of indeterminate equations

4.4.4. Examples of second-degree equations in two variables

Notes from the history of integers

5.1. Fundamental property of algebraic fractions

5.1.1. Definition of an algebraic fraction

5.1.2. Fundamental property of an algebraic fraction. Reducing fractions

5.1.3. Bringing algebraic fractions to a common denominator

Additional exercises for Section 5.1

5.2. Addition and subtraction of algebraic fractions

5.2.1. Adding and subtracting fractions with like denominators

5.2.2. Adding and subtracting fractions with unlike denominators

5.2.3. Representing a given fraction as a sum of several fractions

Additional exercises for Section 5.2

5.3. Multiplication and division of algebraic fractions

5.3.1. Multiplication of fractions

5.3.2. Raising fractions to a power

5.3.3. Division of fractions

Additional exercises for Section 5.3

5.3.4. Transforming rational fractions

5.4. Inverse proportionality, its graph and properties

5.4.1. Inverse proportionality

5.4.2. Graph of inverse proportionality

Reference materials for the Grade 6–7 algebra course

Review exercises

Sample tests

Answers