Theory of combinations. Newton’s binomial Permutations. Factorial. Arrangements. Combinations. Newton’s binomial. Binomial coefficients. Pascal’s triangle. Properties of binomial coefficients. By the general name “combinations” we call three kinds of combinations, composed from some number of different elements, belonging to the same set ( for instance, letters of an alphabet, books of a library, cars […]

## Logarithms

Logarithms Logarithm. Main logarithmic identity. P roperties of logarithms. Common logarithm. Natural logarithm. A logarithm of a positive number N to the base b ( b > 0, b1 ) is called an exponent of a power x , to which b must be raised to receive N . The designation: & nbsp; &n bsp; This record […]

## Arithmetic and geometric progressions

Arithmetic and geometric progressions Sequences. Numerical sequences. General term of numerical sequence. Arithmetic progression. Geometric progression. Infinitely decreasing geometric progression. Converting of repeating decimal to vulgar fraction. Sequences. Let’s consider the series of natural numbers: 1, 2, 3, … , n – 1, n , … .If to replace each natural number n in this […]

## Proving and solving of inequalities

Proving and solving of inequalities Proving of inequalities. Basic methods. Solving of inequalities. Equivalent inequalities. Method of intervals. Double inequality. Systems of simultaneous inequalities. Proving of inequalities. There are some ways to prove inequalities. We’ll consider them to prove the inequality: where a – a positive number . 1). Using of the known or […]

## Inequalities: common information

Inequalities: common information Inequality. Signs of inequalities. Identical inequality. Strict inequality. Non-strict inequality. Solving of inequality or system of simultaneous inequalities. Main properties of inequalities. Some important inequalities. Two expressions ( numerical or literal ones ), connected by one of the signs: “more” (>) , “less” ( “less than or equal to” (≤) form an inequality […]

## Mathematical induction

Mathematical induction Assume it’s necessary to prove a statement ( formula, property etc.), depending on a natural number n . If : 1) this statement is valid for some natural number n0 , 2) from validity of this statement at n = k its validity follows at n = k + 1 […]

## Principles of vector calculus

Principles of vector calculus Vectors. Opposite vectors. Zero vector. Length (modulus) of vector. Collinear vectors. Coplanar vectors. Equality of vectors. Parallel transfer of vectors. Addition of vectors. Subtraction of vectors. Laws of addition of vectors. Laws of multiplication of vector by a number. Scalar product of vectors. Angle between non-zero vectors. Scalar square. Properties of […]

## Factoring of a quadratic trinomial

Factoring of a quadratic trinomial Each quadratic trinomial ax2 + bx+ c can be resolved to factors of the first degree by the next way. Solve the quadratic equation ax2 + bx+ c = 0 . If x1 and x2 are the roots of this equation, then ax2 + bx+ c = a ( x […]

## Properties of roots of a quadratic equation. Viete’s theorem

Properties of roots of a quadratic equation. Viete’s theorem Roots of quadratic equation. Discriminant. Viete’s theorem. The formula shows, that the three cases are possible: 1) b 2 – 4 a c > 0 , then two roots are different real numbers; 2) […]

## Solution of a quadratic equation

Solution of a quadratic equation Formulas for solution of non-reduced and reduced quadratic equation. In general case of a non-reduced quadratic equation: ax 2 + bx + c = 0 , its roots are found by the formula: If to divide all terms of a non-reduced quadratic equation by a (is it possible […]