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Design

Information about the dimensions and weight of the device, presented in different units of measurement. Used materials, offered colors, certificates.

SIM card

The SIM card is used in mobile devices to store data that certifies the authenticity of mobile service subscribers.

Mobile networks

A mobile network is a radio system that allows multiple mobile devices to communicate with each other.

Mobile technology and data rates

Communication between devices in mobile networks is carried out using technologies that provide different data rates.

Operating system

An operating system is the system software that controls and coordinates the operation of hardware components on a device.

SoC (System on a Chip)

A system on a chip (SoC) integrates all the major hardware components of a mobile device into a single chip.

Built-in memory

Each mobile device has built-in (non-removable) fixed memory.

Memory cards

Memory cards are used in mobile devices to increase the storage space for data.

Screen

The screen of a mobile device is characterized by its technology, resolution, pixel density, diagonal length, color depth, etc.

Sensors

Different sensors perform different quantitative measurements and convert physical metrics into signals that can be recognized by the mobile device.

Rear camera

The main camera of a mobile device is usually located on its rear panel and can be combined with one or more additional cameras.

Front-camera

Smartphones have one or more front cameras of different designs - pop-up camera, PTZ camera, notch or hole in the display, camera under the display.

Audio

Information about the type of speakers and audio technology supported by the device.

Radio

The radio of the mobile device is a built-in FM receiver.

Locating

Information about the navigation and positioning technologies supported by the device.

Wi-Fi

Wi-Fi is a technology that enables wireless communication for transferring data over short distances between various devices.

Bluetooth

Bluetooth is a standard for secure wireless transfer of data between different types of devices over short distances.

USB

USB (Universal Serial Bus) is an industry standard that allows different electronic devices to exchange data.

Headphone jack

This is an audio connector, which is also called an audio connector. The most widely used standard in mobile devices is the 3.5mm headphone jack.

Connecting devices

Information about other important connection technologies supported by the device.

Browser

A web browser is a software application for accessing and viewing information on the Internet.

Video file formats / codecs

Mobile devices support different video file formats and codecs, which respectively store and encode / decode digital video data.

Battery

Mobile device batteries differ in their capacity and technology. They provide the electrical charge required for their function.

Mathematical expressions (formulas) abbreviated multiplication(the square of the sum and the difference, the cube of the sum and the difference, the difference of the squares, the sum and the difference of the cubes) are extremely irreplaceable in many areas of the exact sciences. These 7 symbolic notations are irreplaceable for simplifying expressions, solving equations, multiplying polynomials, canceling fractions, solving integrals and much more. This means that it will be very useful to understand how they are obtained, what they are for, and most importantly, how to remember them and then apply them. Then applying abbreviated multiplication formulas in practice, the most difficult thing will be to see what is NS and what do u have. Obviously, there are no restrictions for a and b no, which means it can be any numeric or literal expressions.

And so they are:

The first x 2 - at 2 = (x - y) (x + y).To calculate difference of squares two expressions must be multiplied by the differences of these expressions by their sums.

The second (x + y) 2 = x 2 + 2xy + y 2... To find the square of the sum two expressions, you need to add the double product of the first expression to the second plus the square of the second expression to the square of the first expression.

Third (x - y) 2 = x 2 - 2xy + y 2... To calculate squared difference two expressions, you need to subtract the double product of the first expression by the second plus the square of the second expression from the square of the first expression.

Fourth (x + y) 3 = x 3 + 3x 2 y + 3x 2 + y 3. To calculate cube sum two expressions, you need to add to the cube of the first expression the triple product of the square of the first expression by the second plus triple the product of the first expression by the square of the second plus the cube of the second expression.

Fifth (x - y) 3 = x 3 - 3x 2 y + 3x 2 - at 3... To calculate difference cube of two expressions, it is necessary to subtract from the cube of the first expression the triple product of the square of the first expression by the second plus triple the product of the first expression by the square of the second minus the cube of the second expression.

Sixth x 3 + at 3 = (x + y) (x 2 - xy + y 2) To calculate sum of cubes two expressions, you need to multiply the sums of the first and second expressions by the incomplete square of the difference between these expressions.

Seventh x 3 - at 3 = (x - y) (x 2 + xy + y 2) To perform a calculation difference cubes two expressions, the difference between the first and second expressions must be multiplied by the incomplete square of the sum of these expressions.

It is not difficult to remember that all formulas are applied to perform calculations and in the opposite direction (from right to left).

The existence of these regularities was discovered about 4 thousand years ago. They were widely used by the inhabitants of ancient Babylon and Egypt. But in those times they were expressed verbally or geometrically and did not use letters in the calculations.

Let's analyze sum square proof(a + b) 2 = a 2 + 2ab + b 2.

The first this mathematical pattern proved by the ancient Greek scientist Euclid, who worked in Alexandria in the 3rd century BC, he used for this a geometric method of proving the formula, since the scientists of ancient Greece did not use letters to denote numbers. They widely used not “a 2”, but “a square on a segment a”, not “ab”, but “a rectangle enclosed between segments a and b”.

Abbreviated multiplication formulas.

Study of abbreviated multiplication formulas: the square of the sum and the square of the difference of two expressions; difference of squares of two expressions; the cube of the sum and the cube of the difference of two expressions; sum and difference of cubes of two expressions.

Application of abbreviated multiplication formulas when solving examples.

To simplify expressions, factorize polynomials, and bring polynomials to a standard form, abbreviated multiplication formulas are used. Abbreviated multiplication formulas need to be known by heart.

Let a, b R. Then:

1. The square of the sum of the two expressions is the square of the first expression plus twice the product of the first expression by the second plus the square of the second expression.

(a + b) 2 = a 2 + 2ab + b 2

2. The squared difference of the two expressions is the square of the first expression minus twice the product of the first expression by the second plus the square of the second expression.

(a - b) 2 = a 2 - 2ab + b 2

3. Difference of squares two expressions is equal to the product of the difference between these expressions and their sum.

a 2 - b 2 = (a -b) (a + b)

4. Sum cube two expressions is equal to the cube of the first expression plus three times the square of the first expression and the second plus three times the product of the first expression and the square of the second plus the cube of the second expression.

(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3

5. Difference cube two expressions is equal to the cube of the first expression minus three times the square of the first expression and the second plus three times the product of the first expression and the square of the second minus the cube of the second expression.

(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3

6. Sum of cubes two expressions is equal to the product of the sum of the first and second expressions by the incomplete square of the difference of these expressions.

a 3 + b 3 = (a + b) (a 2 - ab + b 2)

7. Difference of cubes two expressions is equal to the product of the difference of the first and second expressions by the incomplete square of the sum of these expressions.

a 3 - b 3 = (a - b) (a 2 + ab + b 2)

Application of abbreviated multiplication formulas when solving examples.

Example 1.

Calculate

a) Using the formula for the square of the sum of two expressions, we have

(40 + 1) 2 = 40 2 + 2 40 1 + 1 2 = 1600 + 80 + 1 = 1681

b) Using the formula for the square of the difference of two expressions, we get

98 2 = (100 - 2) 2 = 100 2 - 2 100 2 + 2 2 = 10000 - 400 + 4 = 9604

Example 2.

Calculate

Using the formula for the difference between the squares of the two expressions, we get

Example 3.

Simplify expression

(x - y) 2 + (x + y) 2

We use the formulas for the square of the sum and the square of the difference of two expressions

(x - y) 2 + (x + y) 2 = x 2 - 2xy + y 2 + x 2 + 2xy + y 2 = 2x 2 + 2y 2

Abbreviated multiplication formulas in one table:

(a + b) 2 = a 2 + 2ab + b 2
(a - b) 2 = a 2 - 2ab + b 2
a 2 - b 2 = (a - b) (a + b)
(a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3
(a - b) 3 = a 3 - 3a 2 b + 3ab 2 - b 3
a 3 + b 3 = (a + b) (a 2 - ab + b 2)
a 3 - b 3 = (a - b) (a 2 + ab + b 2)