Abbreviated multiplication formulas.
Studying abbreviated multiplication formulas: the square of the sum and the square of the difference of two expressions; difference of squares of two expressions; cube of the sum and cube of the difference of two expressions; sums and differences of cubes of two expressions.
Application of abbreviated multiplication formulas when solving examples.
To simplify expressions, factor polynomials, reduce polynomials to standard view 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 two expressions is equal to the square of the first expression plus twice the product of the first expression and the second plus the square of the second expression.
(a + b) 2 = a 2 + 2ab + b 2
2. The square of the difference of two expressions is equal to the square of the first expression minus twice the product of the first expression and 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 of these expressions and their sum.
a 2 - b 2 = (a -b) (a+b)
4. Cube of sum two expressions is equal to the cube of the first expression plus triple the product of the square of the first expression and the second plus triple 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 triple the product of the square of the first expression and the second plus triple 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 and 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 obtain
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 of the squares of two expressions, we get
Example 3.
Simplify an expression
(x - y) 2 + (x + y) 2
Let's 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)
Mathematical expressions (formulas) abbreviated multiplication(square of sum and difference, cube of sum and difference, difference of squares, sum and difference of cubes) are extremely irreplaceable in many areas of the exact sciences. These 7 symbolic entries are irreplaceable when simplifying expressions, solving equations, multiplying polynomials, abbreviating fractions, decision integrals and much more. This means it will be very useful to understand how they are obtained, why they are needed, 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 X and what do you have. Obviously, there are no restrictions for a And b no, which means it can be any numeric or alphabetic expression.
And so here they are:
First x 2 - at 2 = (x - y) (x+y).To calculate difference squares two expressions, you need to multiply the differences of these expressions by their sums.
Second (x + y) 2 = x 2 + 2xy + y 2. To find square of the sum two expressions, you need to add double the square of the first expression work the first expression by the second plus the square of the second expression.
Third (x - y) 2 = x 2 - 2xy + y 2. To calculate squared difference two expressions are needed from the square of the first expression take away twice the product of the first expression and the second plus the square of the second expression.
Fourth (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + at 3. To calculate cube amounts 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 the triple 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 + 3xy 2 - at 3. To calculate difference cube 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 the triple product of the first expression by the square of the second minus the cube of the second expression.
Sixth x 3 + y 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 of these expressions.
Seventh x 3 - at 3 = (x - y) (x 2 + xy + y 2) To perform the calculation differences of cubes two expressions, you need to multiply the difference of the first and second expressions by the incomplete square of the sum of these expressions.
It is not difficult to remember that all formulas are used to perform calculations in the opposite direction (from right to left).
The existence of these patterns was known about 4 thousand years ago. They were widely used by the inhabitants of ancient Babylon and Egypt. But in those eras they were expressed verbally or geometrically and did not use letters in calculations.
Let's sort it out square proof amounts (a + b) 2 = a 2 +2ab +b 2.
First this mathematical pattern Proved by the ancient Greek scientist Euclid, who worked in Alexandria in the 3rd century BC, he used a geometric method to prove the formula, since the scientists of ancient Hellas did not use letters to denote numbers. They universally used not “a 2”, but “ square on segment a”, not “ab”, but “ rectangle, enclosed between segments a and b.”
Information about the make, model, and alternative names of the specific device, if available.
Design
Information about the dimensions and weight of the device, presented in different units of measurement. Materials used, colors offered, certificates.
| Width Width information - refers to the horizontal side of the device in its standard orientation during use. | 218 mm (millimeters) 21.8 cm (centimeters) 0.72 ft (feet) 8.58 in (inches) |
| Height Height information - refers to the vertical side of the device in its standard orientation during use. | 126 mm (millimeters) 12.6 cm (centimeters) 0.41 ft (feet) 4.96 in (inches) |
| Thickness Information about the thickness of the device in different units measurements. | 7.8 mm (millimeters) 0.78 cm (centimeters) 0.03 ft (feet) 0.31 in (inches) |
| Weight Information about the weight of the device in different units of measurement. | 356 g (grams) 0.78 lbs 12.56 oz (ounces) |
| Volume The approximate volume of the device, calculated based on the dimensions provided by the manufacturer. Refers to devices with the shape of a rectangular parallelepiped. | 214.25 cm³ (cubic centimeters) 13.01 in³ (cubic inches) |
| Colors Information about the colors in which this device is offered for sale. | Black |
| Materials for making the case Materials used to make the device body. | Aluminium alloy |
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.
| GSM GSM (Global System for Mobile Communications) is designed to replace the analogue mobile network (1G). For this reason, GSM is often called a 2G mobile network. It is improved by the addition of GPRS (General Packet Radio Services), and later EDGE (Enhanced Data rates for GSM Evolution) technologies. | GSM 850 MHz GSM 900 MHz GSM 1800 MHz GSM 1900 MHz |
| CDMA CDMA (Code-Division Multiple Access) is a channel access method used in communications in mobile networks. Compared to other 2G and 2.5G standards like GSM and TDMA, it provides more high speeds data transfer and the ability to connect more consumers at the same time. | CDMA 800 MHz |
| W-CDMA W-CDMA (Wideband Code Division Multiple Access) is an air interface used by 3G mobile networks, and is one of the three main air interfaces of UMTS, along with TD-SCDMA and TD-CDMA. It provides even higher data transfer speeds and the ability to connect more consumers at the same time. | W-CDMA 2100 MHz |
| TD-SCDMA TD-SCDMA (Time Division Synchronous Code Division Multiple Access) is a 3G mobile network standard. It is also called UTRA/UMTS-TDD LCR. It was developed as an alternative to the W-CDMA standard in China by the Chinese Academy of Telecommunications Technology, Datang Telecom and Siemens. TD-SCDMA combines TDMA and CDMA. | TD-SCDMA 1880-1920 MHz TD-SCDMA 2010-2025 MHz |
| LTE LTE (Long Term Evolution) is defined as a fourth generation (4G) technology. It is developed by 3GPP based on GSM/EDGE and UMTS/HSPA to increase the capacity and speed of wireless mobile networks. The subsequent technology development is called LTE Advanced. | LTE 1800 MHz LTE 2100 MHz LTE-TDD 1900 MHz (B39) LTE-TDD 2300 MHz (B40) LTE-TDD 2500 MHz (B41) LTE-TDD 2600 MHz (B38) |
Mobile communication technologies and data transfer speeds
Communication between devices on mobile networks is carried out using technologies that provide different data transfer rates.
Operating system
An operating system is a system software that manages and coordinates the operation of hardware components in a device.
SoC (System on Chip)
A system on a chip (SoC) includes all the most important hardware components of a mobile device on one chip.
| SoC (System on Chip) A system on a chip (SoC) integrates various hardware components, such as a processor, graphics processor, memory, peripherals, interfaces, etc., as well as the software necessary for their operation. | MediaTek Helio X20 (MT6797) |
| Technological process Information about technological process, on which the chip is made. Nanometers measure half the distance between elements in the processor. | 20 nm (nanometers) |
| Processor (CPU) The primary function of a mobile device's processor (CPU) is to interpret and execute instructions contained in software applications. | 2x 2.3 GHz ARM Cortex-A72, 4x 1.85 GHz ARM Cortex-A53, 4x 1.4 GHz ARM Cortex-A53 |
| Processor size The size (in bits) of a processor is determined by the size (in bits) of the registers, address buses, and data buses. 64-bit processors have higher performance compared to 32-bit processors, which in turn are more powerful than 16-bit processors. | 64 bit |
| Instruction Set Architecture Instructions are commands with which the software sets/controls the operation of the processor. Information about the instruction set (ISA) that the processor can execute. | ARMv8-A |
| Number of processor cores The processor core performs program instructions. There are processors with one, two or more cores. Having more cores increases performance by allowing multiple instructions to be executed in parallel. | 10 |
| CPU clock speed The clock speed of a processor describes its speed in terms of cycles per second. It is measured in megahertz (MHz) or gigahertz (GHz). | 2300 MHz (megahertz) |
| Graphics Processing Unit (GPU) The Graphics Processing Unit (GPU) handles calculations for various 2D/3D graphics applications. IN mobile devices it is used most often by games, consumer interface, video applications, etc. | ARM Mali-T880 MP4 |
| Number of cores GPU Like a CPU, a GPU is made up of several working parts called cores. They handle graphics calculations for various applications. | 4 |
| GPU clock speed Running speed is the clock speed of the GPU, measured in megahertz (MHz) or gigahertz (GHz). | 780 MHz (megahertz) |
| Volume random access memory(RAM) Random Access Memory (RAM) in use operating system and all installed applications. Data stored in RAM is lost after the device is turned off or restarted. | 4 GB (gigabytes) |
| Type of random access memory (RAM) Information about the type of random access memory (RAM) used by the device. | LPDDR3 |
| Number of RAM channels Information about the number of RAM channels that are integrated into the SoC. More channels mean higher data rates. | Dual channel |
| RAM frequency The frequency of RAM determines its operating speed, more specifically, the speed of reading/writing data. | 800 MHz (megahertz) |
Built-in memory
Each mobile device has built-in (non-removable) memory with a fixed capacity.
Memory cards
Memory cards are used in mobile devices to increase the storage capacity for storing data.
Screen
The screen of a mobile device is characterized by its technology, resolution, pixel density, diagonal length, color depth, etc.
| Type/technology One of the main characteristics of the screen is the technology by which it is made and on which the quality of the information image directly depends. | JDI IPS |
| Diagonal For mobile devices, screen size is expressed by the length of its diagonal, measured in inches. | 8.4 in (inches) 213.36 mm (millimeters) 21.34 cm (centimeters) |
| Width Approximate screen width | 7.12 in (inches) 180.93 mm (millimeters) 18.09 cm (centimeters) |
| Height Approximate screen height | 4.45 in (inches) 113.08 mm (millimeters) 11.31 cm (centimeters) |
| Aspect Ratio The ratio of the dimensions of the long side of the screen to its short side | 1.6:1 16:10 |
| Permission Screen resolution shows the number of pixels vertically and horizontally on the screen. More a high resolution means sharper detail in the image. | 2560 x 1600 pixels |
| Pixel Density Information about the number of pixels per centimeter or inch of the screen. Higher density allows information to be displayed on the screen with clearer detail. | 359 ppi (pixels per inch) 141 ppcm (pixels per centimeter) |
| Color depth Screen color depth reflects the total number of bits used for color components in one pixel. Information about the maximum number of colors that the screen can display. | 24 bit 16777216 flowers |
| Screen area Approximate percentage of screen area occupied by the screen on the front of the device. | 74.73% (percent) |
| Other characteristics Information about other screen features and characteristics. | Capacitive Multi-touch |
| Display manufacturer - Japan Display Inc. OGS (One Glass Solution) |
Sensors
Different sensors perform different quantitative measurements and convert physical indicators into signals that a mobile device can recognize.
Rear camera
The main camera of a mobile device is usually located on its back panel and may be combined with one or more secondary cameras.
| Sensor type Information about the camera sensor type. Some of the most widely used types of sensors in mobile device cameras are CMOS, BSI, ISOCELL, etc. | CMOS (complementary metal-oxide semiconductor) |
| Svetlosila | f/2.2 |
| Flash type The rear (rear) cameras of mobile devices mainly use led flashes. They can be configured with one, two or more light sources and vary in shape. | LED |
| Image Resolution | 4160 x 3120 pixels 12.98 MP (megapixels) |
| Video resolution Information about the maximum video resolution that the camera can record. | 1920 x 1080 pixels 2.07 MP (megapixels) |
| Video recording speed (frame rate) Information about the maximum recording speed (frames per second, fps) supported by the camera at maximum resolution. Some of the most basic video recording speeds are 24 fps, 25 fps, 30 fps, 60 fps. | 30fps (frames per second) |
| Characteristics Information about additional software and hardware features of the rear (rear) camera. | Autofocus Continuous shooting Digital zoom Geographical tags Panoramic photography HDR shooting Touch Focus Face recognition White Balance Adjustment ISO Setting Exposure compensation Self-timer Scene Selection Mode |
Front-camera
Smartphones have one or more front cameras of various designs - a pop-up camera, a rotating camera, a cutout or hole in the display, an under-display camera.
| Svetlosila F-stop (also known as aperture, aperture, or f-number) is a measure of the size of a lens's aperture, which determines the amount of light entering the sensor. The lower the f-number, the larger the aperture and the more light reaches the sensor. Typically the f-number is specified to correspond to the maximum possible aperture of the aperture. | f/2.2 |
| Image Resolution One of the main characteristics of cameras is resolution. It represents the number of horizontal and vertical pixels in an image. For convenience, smartphone manufacturers often list resolution in megapixels, indicating the approximate number of pixels in millions. | 3264 x 2448 pixels 7.99 MP (megapixels) |
Audio
Information about the type of speakers and audio technologies supported by the device.
Radio
The radio of the mobile device is a built-in FM receiver.
Location determination
Information about the navigation and location technologies supported by your device.
WiFi
Wi-Fi is a technology that provides wireless communication for transmitting data over close distances between various devices.
Bluetooth
Bluetooth is a standard for secure wireless data transfer between various devices of different types 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, also called an audio jack. The most widely used standard in mobile devices is the 3.5mm headphone jack.
Connecting devices
Information about other important connection technologies supported by your 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 from each other in their capacity and technology. They provide the electrical charge necessary for their functioning.
| Capacity A battery's capacity indicates the maximum charge it can hold, measured in milliamp-hours. | 4500 mAh (milliamp-hours) |
| Type The type of battery is determined by its structure and, more precisely, the chemicals used. Exist different types batteries, with lithium-ion and lithium-ion polymer batteries most often used in mobile devices. | Li-polymer |
| Adapter output power Power Information electric current(measured in amperes) and electrical voltage(measured in volts) supplied Charger (output power). Higher power output ensures faster battery charging. | 5 V (volts) / 2 A (amps) |
| Characteristics Information about some additional characteristics device battery. | Fixed |
| Battery life - up to 13 hours |
Loading...
