What does the difference between numbers and execution mean. Subtraction of numbers

The difference or subtraction of integers is directly related to the topic of addition of integers. After all, knowing the sum and one of the terms, you can find the second term. Consider an example:

We have 10 apples in the basket. The first time 2 apples were added to the basket, how many apples were added to the basket the second time to end up with 10 apples?
Let x be the number of apples added a second time. If we add two apples to x, we get 10 apples. Mathematically, the entry will look like this:

to find the variable x, you need to remove 2 apples from the basket or subtract one known term 2 from the sum 10.

That is, the variable x=8.

Definition:
The difference of two integers is the integer that, when added to the subtrahend, gives the minuend.

The difference between integers a and b is denoted as a-b.

Differencea-b is the sum of the numbersa and opposite numberb.
a-b=a+(-b)

where b and –b are opposite numbers.

Example:
5-2=5+(-2)=3

Subtraction of positive integers in examples.

Example:
Subtract from the integer 12 the number 5.

Decision:
According to the rule of difference, we must replace the subtracted 5 with the opposite number, that is, -5 and execute.

Example:
From the number 37, subtract the number 56.

Decision:
It is necessary to replace the subtracted number 56 with the opposite number, that is, the number -56 and perform the addition of integers with different signs.

37-56=37+(-56)=-21

Example:
Subtract 7 from -4.

Decision:
We replace the subtracted number 7 with the opposite number -7 and add from according to the rule

4-7=-4+(-7)=-11

Subtraction of negative integers in examples.

Example:
Find the difference between the numbers 6 and -8.

Decision:
According to the rule of difference, you need to replace the subtracted -8 with the opposite number +8 or 8 and calculate the sum of integers. We get:

Subtract -10 from the integer -14.
It is necessary to replace the subtracted -10 with the opposite number +10 or 10 according to the rule for subtracting integers and then perform the addition.

14-(-10)=-14+10=-4

Subtract zero from integers.

If you subtract zero from an integer, then the number does not change..

Consider an example:
3-0=3+0=3

a-0=a

If we subtract zero from zero, we get zero.

Subtraction of identical integers.

Consider the problem:
Misha received 2 sweets from his mother and he immediately treated his friend Sasha with two sweets. How many sweets does Misha have left?

Decision:
Misha received 2 candies and gave away 2 candies, mathematically it can be written as follows:

Answer: Misha has 0 candies left.

That is, if you do Subtracting equal numbers results in zero.

Checking the result of subtraction.

How to check if you have found the difference of two integers correctly?
The answer is simple, it lies in the very definition of the difference of two integers. Need add the difference with the subtrahend, we get the minuend. The verbal formula would look like this:

Difference+Subtracted=Reduced

Example:
19-5=14

19 is our reduced;
5 - subtracted;
14 - difference.

Let's check:
We add the minuend to the difference, if the subtraction was done correctly, we get the minuend.

Another example:
Perform a subtraction test 12-23=-11

12 - reduced;
23 - subtracted;
-11 - difference.

Let's check the subtraction:
Difference+Subtracted=Reduced

To subtract means to subtract one number from another.

Subtraction is an operation in which a smaller number is subtracted from a larger one. When subtracting integers, the larger number is reduced by as many units as there are in the smaller one. Subtracting one number from another means turn down one number to another, so there is a subtraction the reverse action of addition.

In subtraction, two given numbers are called reduced and subtracted , and the desired - difference .

A smaller number is called a larger number, from which another is subtracted. It decreases with subtraction.

The subtracted is the smaller number that is subtracted from the larger one.

The difference is the output obtained from subtraction. The difference determines how one number is greater than another or shows the difference between two numbers.

subtraction sign. The operation of subtraction is indicated by the - (minus) sign.

Single digit subtraction

To indicate that 6 must be subtracted from 9, these numbers are written side by side, separating them with a - (minus) sign:

The difference between these numbers will be 3, and the course of the calculation is expressed verbally:

nine minus six equals three.

In writing:

A larger number 9 will be reduced, a smaller number 6 will be subtracted, the number 3 will be the remainder.

Subtraction methods

There are two ways to subtract one number from another:

or you can subtract as many units from the larger number as there are in the smaller one. So, subtracting 6 from 9 means subtracting 6 from 9. The number 3 will be the desired remainder;

or you can add one to a smaller number until you get a larger number. So, subtracting 6 from 9, we add 3 units to 6. The number of units that must be added to the smaller number to equalize it with the larger one determines the difference. A smaller number with a difference should equal a larger number, therefore, the smaller number and the difference are terms, and the larger one is their sum. Based on this another definition of subtraction:

Subtraction is such an operation in which, given the sum and one term, another term is found.

In this case the given amount is the minuend, the given term is the deductible, and the claimand I difference- another term.

Multi-digit subtraction

Subtraction of multi-digit numbers is based on the property of numbers, according to which subtracting a number is the same as subtracting all its parts. From this property it can be seen that subtracting some number is the same as subtracting successively all its units, tens, hundreds, etc. To indicate that 3517 must be subtracted from the number 7228, they write:

and subtract separately units from units, tens from tens, etc.

To facilitate subtraction, they sign a smaller number under a large one so that units of the same order are in the same vertical column, draw a line, put a subtraction sign on the left - and sign the difference under the line.

The course of calculation is expressed verbally:

Starting subtraction with simple units: 8 minus 7 is 1; signed under units 1.

Subtract tens: 2 without 1 gives 1, we sign under tens 1.

Subtract hundreds. Five cannot be subtracted from 2, so we take one from the next higher order (thousands), which we denote by putting a dot over 7. Each order unit contains 10 units of the next lower order. Adding these 10 units to 2, we get 12; 12 without 5 is 7, we sign under hundreds 7. When one is taken from a higher order, this is indicated by putting a dot over the order from which they occupy.

Subtract thousands. Instead of 7, only 6 thousand remained, for one was taken. 6 minus 3 is 3; sign under thousands 3.

The progress of the calculation is expressed in writing:

Example. Subtract 6025 from 17004.

5 cannot be subtracted from 4. We borrow one from tens, the next highest order, but there are no ones in this order; we borrow from hundreds, and there are no hundreds; we borrow from thousands and denote this with a dot above the number 7.

The unit of the fourth has 10 units of the third order. Taking one of them for tens, we leave them in hundreds only 9. Adding 10 to 4, we have 14.

Subtracting, we get:

for units 14 - 5 = 9

for tens 9 - 2 = 7

for hundreds 9 - 0 = 9

for thousands 6 - 6 = 0

For tens of thousands, we have 1, because we transfer this figure of the reduced to the difference without change.

The course of calculation will be expressed in writing:

From the previous examples, we deduce subtraction rules:

To make the subtraction of integers, you need to sign the subtrahend under the minuend so that units of the same order are in the same vertical column, draw a line, under which you sign the difference.

Subtraction must begin with simple units, that is, from the first column, and then, moving to the next columns from the right hand to the left, subtract tens from tens, hundreds from hundreds, etc.

If the digit of the subtracted is less than the digit of the reduced, the difference is signed in the same column; if the digits are equal, the difference will be zero. If the digit of the subtrahend is greater than the corresponding digit of the reduced, take one from the next order of the reduced, marking this with a dot placed above the figure from which it is occupied, apply 10 to the digit of the reduced and subtract. The number with a dot is considered one less.

If, when subtracting, the digit of the minuend, from which it is occupied, will be 0, followed by zeros in the minuend, then they occupy the first significant digit, putting dots above it and all intermediate zeros. A digit with a dot is counted as one less, and zeros with a dot are counted as 9.

The subtraction is continued until the total difference is obtained.

The extra digits of the minuend are transferred to the difference.

Relationship between data and desired subtractions

From example 9 - 6 = 3, it can be seen that

The minuend is equal to the subtrahend, added to the difference: 9 = 6 + 3.

Subtrahend equals minuend without difference: 6 = 9 - 3.

The difference is equal to the minuend without the subtrahend: 3 = 9 - 6.

Arithmetic addition. The difference between a number and the nearest larger unit is called arithmetic complement. So, the arithmetic complements of the numbers 7, 79, 983 will be the numbers:

10 - 7 = 3
100 - 79 = 21
1000 - 983 = 17

Arithmetic addition is sometimes used to facilitate arithmetic calculations.

There are four basic arithmetic operations: addition, subtraction, multiplication and division. They are the basis of mathematics, with their help all other, more complex calculations are performed. Addition and subtraction are the simplest of them and are mutually opposite. But with the terms used in addition, we often encounter in life.

We are talking about “combining efforts” when trying to jointly obtain the desired result, about “components of achieved success”, etc. The names associated with subtraction remain within the bounds of mathematics, rarely appearing in everyday speech. Therefore, the words "subtracted", "reduced", "difference" are less common. The rule of finding each of these components can be applied only if the meaning of these names is understood.

Unlike many scientific terms that have Greek, Latin or Arabic origin, in this case words with Russian roots are used. So it is not difficult to understand their meaning, which means it is easy to remember what is denoted by what term.

If you look closely at the name itself, it becomes noticeable that it is related to the words "different", "difference". From this it can be concluded that what is meant is the established difference between the quantities.

This concept in mathematics means:

the difference between two numbers;
it is a measure of how much one quantity is greater or less than another;
this is the result obtained when subtracting - such a definition is offered by the school curriculum.

Note! If the quantities are equal to each other, then there is no difference between them. So their difference is zero.

What is minuend and subtrahend

As the name suggests, less is what is done less. And you can make the quantity smaller by subtracting a part from it. Thus, a diminished number is a number from which a part is taken away.

Subtracted, respectively, is the number that is subtracted from it.

Minuend		Subtrahend		Difference
18	—	11	=	7
14	—	5	=	9
26	—	22	=	4

Useful video: reduced, subtracted, difference

Rules for finding an unknown element

Having understood the terms, it is easy to establish by which rule each of the elements of subtraction is located.

Since the difference is the result of this arithmetic operation, it is found using this operation, no other rules are required here. But they are there in case the other term of the mathematical expression is unknown.

How to find the minuend

This term, as it was found out, refers to the amount from which the part was subtracted. But if one was subtracted, and the other remained in the end, therefore, the number consists of these two parts. It turns out that you can find the unknown reduced by adding two known elements.

So, in this case, to find the unknown, you should add the subtrahend and the difference:

Likewise in all such cases:

?	–	5	=	9
9	+	5	=	14

?	–	22	=	4
4	+	22	=	26

How to find subtrahend

If the whole consists of two parts (in this case, quantities), then subtracting one of them will result in the second. Thus, to find the unknown subtrahend, it is enough to subtract the difference from the whole instead.

Other similar examples are solved by the same rule.

14	–	?	=	9
14	–	9	=	5

The word difference can be used in many ways. It can also mean a difference in something, for example, opinions, views, interests. In some scientific, medical and other professional fields, this term refers to various indicators, for example, blood sugar levels, atmospheric pressure, weather conditions. The concept of "difference", as a mathematical term, also exists.

In contact with

Arithmetic operations with numbers

The basic arithmetic operations in mathematics are:

addition;
subtraction;
multiplication;
division.

Each result of these actions also has its own name:

sum - the result obtained by adding numbers;
difference - the result obtained by subtracting numbers;
product - the result of multiplying numbers;
quotient is the result of division.

Explaining the concepts of sum, difference, product and quotient in mathematics in a simpler language, we can simply write them down only as phrases:

amount - add;
difference - take away;
product - multiply;
private - share.

Considering definitions, what is the difference of numbers in mathematics, this concept can be denoted in several ways:

And all these definitions are true.

How to find the difference in values

Let us take as a basis the notation of the difference that the school curriculum offers us:

The difference is the result of subtracting one number from another. The first of these numbers, from which the subtraction is carried out, is called the minuend, and the second, which is subtracted from the first, is called the subtrahend.

Once again resorting to the school curriculum, we find a rule for how to find the difference:

To find the difference, subtract the minuend from the minuend.

All clear. But at the same time, we got a few more mathematical terms. What do they mean?

Decreasing is a mathematical number from which it is subtracted and it decreases (becomes smaller).
The subtrahend is the mathematical number that is subtracted from the minuend.

Now it is clear that the difference consists of two numbers, which must be known in order to calculate it. And how to find them, we also use the definitions:

To find the minuend, add the difference to the minuend.
To find the subtrahend, you need to subtract the difference from the minuend.

Mathematical operations with the difference of numbers

Based on the derived rules, we can consider illustrative examples. Mathematics is an interesting science. Here we will take only the simplest numbers for solution. Having learned to subtract them, you will learn how to solve more complex values, three-digit, four-digit, integer, fractional, in powers, roots, others.

Simple examples

Example 1. Find the difference between two values.

20 - decreasing value,

15 - subtracted.

Solution: 20 - 15 = 5

Answer: 5 - the difference in values.

Example 2. Find the minuend.

48 - difference,

32 - subtracted value.

Solution: 32 + 48 = 80

Example 3. Find the value to be subtracted.

7 - difference,

17 - reduced value.

Solution: 17 - 7 = 10

Answer: the subtracted value is 10.

More complex examples

In examples 1-3, actions with simple integers are considered. But in mathematics, the difference is calculated using not only two, but also several numbers, as well as integer, fractional, rational, irrational, etc.

Example 4. Find the difference between three values.

Integer values are given: 56, 12, 4.

56 - decreasing value,

12 and 4 are subtracted values.

The solution can be done in two ways.

Method 1 (consecutive subtraction of subtracted values):

1) 56 - 12 = 44 (here 44 is the resulting difference between the first two values, which will be reduced in the second action);

Method 2 (subtracting two subtracted from the reduced sum, which in this case are called terms):

1) 12 + 4 = 16 (where 16 is the sum of two terms, which will be subtracted in the next step);

2) 56 - 16 = 40.

Answer: 40 is the difference of three values.

Example 5. Find the difference between rational fractional numbers.

Given fractions with the same denominators, where

4/5 - reduced fraction,

3/5 - subtracted.

To complete the solution, you need to repeat the actions with fractions. That is, you need to know how to subtract fractions with the same denominator. How to deal with fractions that have different denominators. They must be able to bring them to a common denominator.

Solution: 4/5 - 3/5 = (4 - 3)/5 = 1/5

Answer: 1/5.

Example 6. Triple the difference of numbers.

But how to execute such an example when you want to double or triple the difference?

Let's go back to the rules:

A double number is a value multiplied by two.
A triple number is a value multiplied by three.
The doubled difference is the difference in values multiplied by two.
A triple difference is the difference in values multiplied by three.

7 - reduced value,

5 - subtracted value.

2) 2 * 3 = 6. Answer: 6 is the difference between the numbers 7 and 5.

Example 7. Find the difference between 7 and 18.

7 - reduced value;

18 - subtracted.

Everything seems to be clear. Stop! Is the subtrahend greater than the minuend?

And again, there is a rule applied for a specific case:

If the subtracted is greater than the minuend, the difference will be negative.

Answer: - 11. This negative value is the difference between the two values, provided that the subtracted value is greater than the reduced one.

Math for Blondes

On the World Wide Web, you can find a lot of thematic sites that will answer any question. In the same way, online calculators for every taste will help you in any mathematical calculations. All the calculations made on them are a great help for the hasty, uninquisitive, lazy. Math for Blondes is one such resource. And we all resort to it, regardless of hair color, gender and age.

At school, we were taught to calculate such actions with mathematical quantities in a column, and later on a calculator. The calculator is also a handy tool. But, for the development of thinking, intellect, outlook and other vital qualities, we advise you to perform arithmetic operations on paper or even in your mind. The beauty of the human body is the great achievement of the modern fitness plan. But the brain is also a muscle that sometimes needs to be pumped. So, without delay, start thinking.

And even if at the beginning of the path the calculations are reduced to primitive examples, everything is ahead of you. And there is a lot to learn. We see that there are many actions with different values in mathematics. Therefore, in addition to the difference, it is necessary to study how to calculate the rest of the results of arithmetic operations:

sum - by adding the terms;
product - by multiplying factors;
quotient - dividing the dividend by the divisor.

Here is some interesting math.

Determining the sum of numbers

sum (lat. summa- total, total number) of numbers is the result of summing these numbers:. In particular, if two numbers and are added together, then

Exercise. Find the sum of numbers:

Answer.

Sum Properties

Associativity:

Based on these properties, we can conclude that the sum does not change from the rearrangement of the places of the terms.

Distributivity with respect to multiplication

Exercise. Find the sum of numbers in a convenient way:

Decision. By the properties of addition, we have

Answer. 1)

When adding large numbers or decimals, column addition is used.

Decision. We add these numbers in a column, for this we write them one under the other, the discharge under the discharge. In the case of decimal fractions, we focus on the fact that the comma of the first number is under the comma of the second. Next, add the numbers standing one under the other, moving from right to left and writing the result under the line of the fraction. If the sum of numbers in one column exceeds ten, then the number of tens is added to the numbers in the next column to the left of this column:

Answer. 1)

The addition of rational fractions is carried out according to the rule

Decision. Calculate the first sum using the rule of addition of rational numbers

The numerator and denominator of the resulting fraction can be reduced by 2, then in the answer we get

To calculate the second sum, we first convert the second term into an improper fraction, for this we multiply the integer part by the denominator and add the resulting number to the numerator. Next, apply the rule of addition of rational fractions

We select the integer part in the resulting fraction, for this we divide the numerator by the denominator with a remainder. We write the resulting quotient in the integer part, and the remainder of the division in the numerator.

Answer. 1) ; 2)

How to Find the Difference of Numbers in Math

Arithmetic operations with numbers

quotient is the result of division.

amount - add;

product - multiply;

The difference between the numbers means how much one of them is greater than the other.

This is the number that is the remainder when two values are minus.

It is the result of one of the four arithmetic operations, which is subtraction.

This is what happens if you subtract the subtrahend from the minuend.

How to find the difference in values

The difference is the result of subtracting one number from another. The first of these numbers, from which the subtraction is carried out, is called the minuend, and the second, which is subtracted from the first, is called the subtrahend.

Once again resorting to the school curriculum, we find a rule for how to find the difference:

Now it is clear that the difference consists of two numbers, which must be known in order to calculate it. And how to find them, we also use the definitions:

Example 3. Find the value to be subtracted.

Solution: 17 - 7 = 10

Integer values are given: 56, 12, 4.

12 and 4 are subtracted values.

Method 1 (consecutive subtraction of subtracted values):

Method 2 (subtracting two subtracted from the reduced sum, which in this case are called terms):

Answer: 40 is the difference of three values.

Example 5. Find the difference between rational fractional numbers.

Given fractions with the same denominators, where

4/5 - reduced fraction,

Solution: 4/5 - 3/5 = (4 - 3)/5 = 1/5

But how to execute such an example when you want to double or triple the difference?

A double number is a value multiplied by two.
A triple number is a value multiplied by three.
The doubled difference is the difference in values multiplied by two.
A triple difference is the difference in values multiplied by three.

2) 2 * 3 = 6. Answer: 6 is the difference between the numbers 7 and 5.

7 - reduced value;

If the subtracted is greater than the minuend, the difference will be negative.

product - by multiplying factors;
quotient - dividing the dividend by the divisor.

The basic arithmetic operations in mathematics are:

Each result of these actions also has its own name:

sum - the result obtained by adding numbers;
product - the result of multiplying numbers;

This is interesting: what is the modulus of a number?

difference - take away;
private - share.

Considering definitions, what is the difference of numbers in mathematics, this concept can be denoted in several ways:

It is the subtraction of one number from another.

Let us take as a basis the notation of the difference that the school curriculum offers us:

Decreasing is a mathematical number from which it is subtracted and it decreases (becomes smaller).
The subtrahend is the mathematical number that is subtracted from the minuend.
To find the minuend, add the difference to the minuend.
To find the subtrahend, you need to subtract the difference from the minuend.

Mathematical operations with the difference of numbers

Solution: 20 - 15 = 5

Solution: 32 + 48 = 80

Answer: the subtracted value is 10.

More complex examples

The solution can be done in two ways.

1) 56 - 12 = 44 (here 44 is the resulting difference between the first two values, which will be reduced in the second action);

1) 12 + 4 = 16 (where 16 is the sum of two terms, which will be subtracted in the next step);

Everything seems to be clear. Stop! Is the subtrahend greater than the minuend?

Math for Blondes

difference - the result obtained by subtracting numbers;

Explaining the concepts of sum, difference, product and quotient in mathematics in a simpler language, we can simply write them down only as phrases:

Difference in mathematics

The difference in mathematics is the result obtained by subtracting two or more numbers from each other.
This is the value that is the result of subtracting two values.
The difference shows the quantitative difference between two numbers.

And all these definitions are true.

To find the difference, subtract the minuend from the minuend.

All clear. But at the same time, we got a few more mathematical terms. What do they mean?

Simple examples

Example 1. Find the difference between two values.

20 - decreasing value,

Answer: 5 - the difference in values.

Example 2. Find the minuend.

32 - subtracted value.

17 - reduced value.

Example 4. Find the difference between three values.

56 - decreasing value,

Example 6. Triple the difference of numbers.

Let's go back to the rules:

7 - reduced value,

5 - subtracted value.

Example 7. Find the difference between 7 and 18.

And again, there is a rule applied for a specific case:

Answer: - 11. This negative value is the difference between the two values, provided that the subtracted value is greater than the reduced one.

sum - by adding the terms;

Here is some interesting math.

1st grade Mathematics. "Sum and Meaning of Sum"

Goals:

To acquaint and form the ability to use the mathematical terms "sum", "value of the sum". Improve your computing skills.

Develop the ability to compare, analyze, generalize. Develop mathematical speech, interest in mathematics.

Cultivate independence, discipline, ability to work in a team.

Equipment: Chalk, board, cards, multimedia installation, presentation.

1. Organization of the class for the lesson.

2. Reporting the topic and objectives of the lesson:

Today in the lesson we will discover and reveal the secrets of mathematics. So, go!

3. Acquaintance with new material.

Guys, do you like fairy tales? What about Walt Disney stories? Now I will read an excerpt from a fairy tale, and you try to guess who it is.

Wake up, friend Owl! - the hare Fat Man shouted merrily. - A new prince has been born!

The good news immediately spread through the forest, and all the forest dwellers hurried to look at the newborn deer. They were touched, looking at how he tries to get up. His legs were still too weak, and he fell all the time.

Who recognized him? This is, indeed, a deer named Bambi. And then one day it was time to introduce him to the forest. From a fairy tale, we all know that Bambi is inquisitive, so he was delighted with everything he saw around.

Let's go with a deer to an unusual "forest-mathematics".

The deer enters the clearing and sees many flowers. But looking closer, he notices that the flowers hold some kind of secret.

Help him solve this mystery.

Look and tell me what you see? What are the different mathematical notations we can make?

Abbreviated multiplication formulas

When calculating algebraic polynomials, to simplify calculations, we use abbreviated multiplication formulas. There are seven such formulas in total. They all need to be known by heart.

It should also be remembered that instead of "a" and "b" in the formulas, there can be both numbers and any other algebraic polynomials.

Difference of squares

Difference of squares two numbers is equal to the product of the difference of these numbers and their sum.

a 2 − b 2 = (a − b)(a + b)

15 2 − 2 2 = (15 − 2)(15 + 2) = 13 17 = 221
9a 2 − 4b 2 with 2 = (3a − 2bc)(3a + 2bc)

sum square

The square of the sum of two numbers is equal to the square of the first number plus twice the product of the first number and the second plus the square of the second number.

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

Note that with this reduced multiplication formula, it is easy to find the squares of large numbers without using a calculator or long multiplication. Let's explain with an example:

Let's decompose 112 into the sum of numbers whose squares we remember well.
112 = 100 + 1
We write the sum of numbers in brackets and put a square over the brackets.
112 2 = (100 + 12) 2
Let's use the sum square formula:
112 2 = (100 + 12) 2 = 100 2 + 2 100 12 + 12 2 = 10,000 + 2,400 + 144 = 12,544

Remember that the square sum formula is also valid for any algebraic polynomials.

(8a + c) 2 = 64a 2 + 16ac + c 2

The square of the difference

The square of the difference between two numbers is equal to the square of the first number minus twice the product of the first and the second plus the square of the second number.

(a − b) 2 = a 2 − 2ab + b 2

It is also worth remembering a very useful transformation:

The formula above is proved by simply expanding the parentheses:

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

The cube of the sum of two numbers is equal to the cube of the first number plus three times the square of the first number times the second plus three times the product of the first times the square of the second plus the cube of the second.

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

How to remember the sum cube

Remembering this "terrible"-looking formula is quite simple.

Learn that "a 3" comes at the beginning.
The two polynomials in the middle have coefficients of 3.
Recall that any number to the zero power is 1. (a 0 = 1, b 0 = 1) . It is easy to see that in the formula there is a decrease in the degree "a" and an increase in the degree "b". You can verify this:
(a + b) 3 = a 3 b 0 + 3a 2 b 1 + 3a 1 b 2 + b 3 a 0 = a 3 + 3a 2 b + 3ab 2 + b 3

Warning!

difference cube

difference cube of two numbers is equal to the cube of the first number minus three times the square of the first number and the second plus three times the product of the first number and the square of the second minus the cube of the second.

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

This formula is remembered like the previous one, but only taking into account the alternation of the signs "+" and "-". Before the first term "a 3" is "+" (according to the rules of mathematics, we do not write it). This means that the next member will be preceded by “-”, then again “+”, etc.

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

Sum of cubes

Not to be confused with the sum cube!

Sum of cubes is equal to the product of the sum of two numbers by the incomplete square of the difference.

a 3 + b 3 = (a + b)(a 2 − ab + b 2)

The sum of cubes is the product of two brackets.

The first parenthesis is the sum of two numbers.
The second bracket is the incomplete square of the difference of numbers. The incomplete square of the difference is called the expression:
(a 2 − ab + b 2)
This square is incomplete, since in the middle, instead of a double product, there is an ordinary product of numbers.

Difference of cubes

Not to be confused with the difference cube!

Difference of cubes is equal to the product of the difference of two numbers by the incomplete square of the sum.

a 3 − b 3 = (a − b)(a 2 + ab + b 2)

Be careful when writing characters.

Application of abbreviated multiplication formulas

It should be remembered that all the formulas above are also used from right to left.

Many examples in textbooks are designed for you to use formulas to assemble the polynomial back.

a 2 + 2a + 1 = (a + 1) 2
(ac − 4b)(ac + 4b) = a 2 c 2 − 16b 2

You can download a table with all the formulas for abbreviated multiplication in the "Cribs" section.

21. The Cube of the Sum and the Cube of the Difference. rules

For any values of a and b, the equality is true

(a + b) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 . (one)

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

A 3 + 2 a 2 b + a b 2 + a 2 b + 2 a b 2 + b 3 =

A 3 + 3 a 2 b + 3 a b 2 + b 3

Since equality (1) is true for any values of a and b,
sum cube formula. If in this formula instead of a and b
then the identity is again obtained.

(5 y 3 + 2 z) 3 = 125 y 9 + 150 y 6 z + 60 y 3 z 2 + 8 z 3 . (2)

Therefore, the sum cube formula reads like this:

the cube of the sum of two expressions is equal to the cube of the first expression
plus three times 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 − 3 a 2 b + 3 a b 2 − b 3 . (3)

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

A 3 − 2 a 2 b + a b 2 − a 2 b + 2 a b 2 − b 3 =

A 3 − 3 a 2 b + 3 a b 2 − b 3

Since equality (3) is true for any values of a and b,
then it is an identity. This identity is called
difference cube formula. If in this formula instead of a and b
substitute some expressions, for example 5 y 3 and 2 z ,
then the identity is again obtained.

(5 y 3 − 2 z) 3 = 125 y 9 − 150 y 6 z + 60 y 3 z 2 − 8 z 3 . (4)

Therefore, the difference cube formula reads as follows:

the cube of the difference of two expressions is equal to the cube of the first expression
minus the triple 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.

Tasks on the topic "Sum Cube and Difference Cube"

Using the sum or difference cube formula, transform the expression
into a standard form polynomial and choose the correct answer.

1) = a 3 - 3 a 2 c + 3 a c 2 - c 3

2) = a 3 − 3 a 2 c + 3 a c 2 + c 3

3) = a 3 − 3 a c 2 + 3 a c 2 − c 3 False. Do not click on an empty field. (x + 2y) 3 =

1) = x 3 + 6 x 2 y + 6 x y 2 + 4 y 3

2) \u003d x 3 + 6 x 2 y + 12 x y 2 + 8 y 3

3) = x 3 + 6 x 2 y + 6 x y 2 + 8 y 3 False. Wrong. Wrong. Do not click on an empty field. Wrong. (3 a − 2 b) 3 =

1) = 27 a 3 - 27 a 2 b + 12 a b 2 - 8 b 3

2) = 27 a 3 - 54 a 2 b + 36 a b 2 - 8 b 3

3) = 27 a 3 − 18 a 2 b + 18 a b 2 − 8 b 3 False. Wrong. Do not click on an empty field. Wrong. (

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