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Open parenthesis is a number. Expanding brackets - Knowledge Hypermarket

Parentheses are used to indicate the order in which actions are performed in numeric and alphabetic expressions, as well as in expressions with variables. It is convenient to pass from an expression with brackets to identically equal expression without brackets. This technique is called parenthesis opening.

To expand brackets means to rid the expression of these brackets.

Another point deserves special attention, which concerns the peculiarities of writing solutions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as equality. For example, after opening the parentheses, instead of the expression
3−(5−7) we get the expression 3−5+7. We can write both of these expressions as the equality 3−(5−7)=3−5+7.

And one more important point. In mathematics, to reduce entries, it is customary not to write a plus sign if it is the first in an expression or in brackets. For example, if we add two positive numbers, for example, seven and three, then we write not +7 + 3, but simply 7 + 3, despite the fact that seven is also a positive number. Similarly, if you see, for example, the expression (5 + x) - know that there is a plus in front of the bracket, which is not written, and there is a plus + (+5 + x) in front of the five.

Bracket expansion rule for addition

When opening brackets, if there is a plus before the brackets, then this plus is omitted along with the brackets.

Example. Open the brackets in the expression 2 + (7 + 3) Before the brackets plus, it means that the characters in front of the numbers in the brackets do not change.

2 + (7 + 3) = 2 + 7 + 3

The rule for expanding brackets when subtracting

If there is a minus before the brackets, then this minus is omitted along with the brackets, but the terms that were in the brackets change their sign to the opposite. The absence of a sign before the first term in parentheses implies a + sign.

Example. Open brackets in expression 2 − (7 + 3)

There is a minus before the brackets, so you need to change the signs before the numbers from the brackets. There is no sign in brackets before the number 7, which means that the seven is positive, it is considered that the + sign is in front of it.

2 − (7 + 3) = 2 − (+ 7 + 3)

When opening the brackets, we remove the minus from the example, which was before the brackets, and the brackets themselves 2 − (+ 7 + 3), and change the signs that were in the brackets to the opposite ones.

2 − (+ 7 + 3) = 2 − 7 − 3

Expanding parentheses when multiplying

If there is a multiplication sign in front of the brackets, then each number inside the brackets is multiplied by the factor in front of the brackets. At the same time, multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus, gives a minus.

Thus, parentheses in products are expanded in accordance with the distributive property of multiplication.

Example. 2 (9 - 7) = 2 9 - 2 7

When multiplying parenthesis by parenthesis, each term of the first parenthesis is multiplied with every term of the second parenthesis.

(2 + 3) (4 + 5) = 2 4 + 2 5 + 3 4 + 3 5

In fact, there is no need to remember all the rules, it is enough to remember only one, this one: c(a−b)=ca−cb. Why? Because if we substitute one instead of c, we get the rule (a−b)=a−b. And if we substitute minus one, we get the rule −(a−b)=−a+b. Well, if you substitute another bracket instead of c, you can get the last rule.

Expand parentheses when dividing

If there is a division sign after the brackets, then each number inside the brackets is divisible by the divisor after the brackets, and vice versa.

Example. (9 + 6) : 3=9: 3 + 6: 3

How to expand nested parentheses

If the expression contains nested brackets, then they are expanded in order, starting with external or internal.

At the same time, when opening one of the brackets, it is important not to touch the other brackets, just rewriting them as they are.

Example. 12 - (a + (6 - b) - 3) = 12 - a - (6 - b) + 3 = 12 - a - 6 + b + 3 = 9 - a + b

to form the ability to open brackets, taking into account the sign in front of the brackets;

  • developing:
    • develop logical thinking, attention, mathematical speech, the ability to analyze, compare, generalize, draw conclusions;
  • educators:
    •  formation of responsibility, cognitive interest in the subject

    During the classes

    I. Organizational moment.

    Check it out buddy
    Are you ready for the lesson?
    Is everything in place? Everything is good?
    Pen, book and notebook.
    Is everyone seated correctly?
    Is everyone watching closely?

    I want to start the lesson with a question for you:

    What do you think is the most valuable thing on earth? (Children's answers.)

    This question has troubled mankind for thousands of years. Here is the answer given by the famous scientist Al-Biruni: “Knowledge is the most excellent possession. Everyone strives for it, but it does not come by itself.”

    Let these words be the motto of our lesson.

    II. Actualization of previous knowledge, skills, skills:

    Verbal counting:

    1.1. What is today's date?

    2. What do you know about the number 20?

    3. And where is this number located on the coordinate line?

    4. Name the number of his reverse.

    5. Name the number opposite to it.

    6. What is the name of the number - 20?

    7. What numbers are called opposites?

    8. What numbers are called negative?

    9. What is the modulus of the number 20? - 20?

    10. What is the sum of opposite numbers?

    2. Explain the following entries:

    a) The ancient mathematician of genius Archimedes was born in 0 287 BC.

    b) The brilliant Russian mathematician N.I. Lobachevsky was born in 1792.

    for the first time Olympic Games took place in Greece in 776.

    d) The first International Olympic Games took place in 1896.

    e) The XXII Olympic Winter Games took place in 2014.

    3. Find out what numbers are spinning on the “math carousel” (all actions are performed orally).

    II. Formation of new knowledge, skills and abilities.

    You have learned how to perform different operations with integers. What are we going to do next? How will we solve examples and equations?

    Let's find the meaning of these expressions

    7 + (3 + 4) = -7 + 7 = 0
    -7 + 3 + 4 = 0

    What is the procedure in 1 example? How much is in brackets? The order of actions in the second example? Result of the first action? What can be said about these expressions?

    Of course, the results of the first and second expressions are the same, so you can put an equal sign between them: -7 + (3 + 4) = -7 + 3 + 4

    What have we done with the brackets? (Lost.)

    What do you think we will do in class today? (Children formulate the topic of the lesson.) In our example, what sign is in front of the brackets. (Plus.)

    And so we come to the next rule:

    If there is a + sign before the brackets, then you can omit the brackets and this + sign, preserving the signs of the terms in brackets. If the first term in brackets is written without a sign, then it must be written with a + sign.

    But what if there is a minus sign in front of the brackets?

    In this case, you need to reason in the same way as when subtracting: you need to add the number opposite to the one being subtracted:

    7 – (3 + 4) = -7 + (-7) = -7 + (-3) + (-4) = -7 – 3 – 4 = -14

    - So, we opened the brackets when there was a minus sign in front of them.

    The rule for expanding brackets when there is a “-” sign in front of the brackets.

    To open the brackets preceded by the - sign, you need to replace this sign with +, changing the signs of all the terms in the brackets to the opposite, and then open the brackets.

    Let's listen to the rules for opening brackets in verses:

    There is a plus in front of the parenthesis.
    He talks about it
    What are you dropping the brackets
    Let all the signs out!
    Before the parenthesis minus strict
    Will block our way
    To remove brackets
    We need to change the signs!

    Yes, guys, the minus sign is very insidious, it is a “watchman” at the gate (brackets), it releases numbers and variables only when they change their “passports”, that is, their signs.

    Why do you need to open parentheses at all? (When there are brackets, there is a moment of some element of incompleteness, some kind of mystery. It's like closed door, behind which is something interesting.) Today we have learned this secret.

    A small digression into history:

    Curly brackets appear in the writings of Vieta (1593). Brackets were widely used only in the first half of the 18th century, thanks to Leibniz and even more so to Euler.

    Fizkultminutka.

    III. Consolidation of new knowledge, skills and abilities.

    Textbook work:

    No. 1234 (open brackets) - orally.

    No. 1236 (open brackets) - orally.

    No. 1235 (find the meaning of the expression) - in writing.

    No. 1238 (simplify the expressions) - work in pairs.

    IV. Summing up the lesson.

    1. Scores are announced.

    2. House. exercise. 39 No. 1254 (a, b, c), 1255 (a, b, c), 1259.

    3. What have we learned today?

    What have you learned?

    And I want to end the lesson with wishes for each of you:

    “Show the ability to mathematics,
    Do not be lazy, but develop daily.
    Multiply, divide, labor, think,
    Don't forget to be friends with mathematics.

    The main function of brackets is to change the order of actions when calculating values. for example, in the numerical expression \(5 3+7\) the multiplication will be calculated first, and then the addition: \(5 3+7 =15+7=22\). But in the expression \(5·(3+7)\), addition in brackets will be calculated first, and only then multiplication: \(5·(3+7)=5·10=50\).


    Example. Expand the bracket: \(-(4m+3)\).
    Decision : \(-(4m+3)=-4m-3\).

    Example. Expand the bracket and give like terms \(5-(3x+2)+(2+3x)\).
    Decision : \(5-(3x+2)+(2+3x)=5-3x-2+2+3x=5\).


    Example. Expand the brackets \(5(3-x)\).
    Decision : We have \(3\) and \(-x\) in the bracket, and five in front of the bracket. This means that each member of the bracket is multiplied by \ (5 \) - I remind you that the multiplication sign between a number and a bracket in mathematics is not written to reduce the size of records.


    Example. Expand the brackets \(-2(-3x+5)\).
    Decision : As in the previous example, the bracketed \(-3x\) and \(5\) are multiplied by \(-2\).

    Example. Simplify the expression: \(5(x+y)-2(x-y)\).
    Decision : \(5(x+y)-2(x-y)=5x+5y-2x+2y=3x+7y\).


    It remains to consider the last situation.

    When multiplying parenthesis by parenthesis, each term of the first parenthesis is multiplied with every term of the second:

    \((c+d)(a-b)=c (a-b)+d (a-b)=ca-cb+da-db\)

    Example. Expand the brackets \((2-x)(3x-1)\).
    Decision : We have a product of brackets and it can be opened immediately using the formula above. But in order not to get confused, let's do everything step by step.
    Step 1. Remove the first bracket - each of its members is multiplied by the second bracket:

    Step 2. Expand the products of the bracket by the factor as described above:
    - the first one first...

    Then the second.

    Step 3. Now we multiply and bring like terms:

    It is not necessary to paint all the transformations in detail, you can immediately multiply. But if you are just learning to open brackets - write in detail, there will be less chance of making a mistake.

    Note to the entire section. In fact, you don't need to remember all four rules, you only need to remember one, this one: \(c(a-b)=ca-cb\) . Why? Because if we substitute one instead of c, we get the rule \((a-b)=a-b\) . And if we substitute minus one, we get the rule \(-(a-b)=-a+b\) . Well, if you substitute another bracket instead of c, you can get the last rule.

    parenthesis within parenthesis

    Sometimes in practice there are problems with brackets nested inside other brackets. Here is an example of such a task: to simplify the expression \(7x+2(5-(3x+y))\).

    To be successful in these tasks, you need to:
    - carefully understand the nesting of brackets - which one is in which;
    - open the brackets sequentially, starting, for example, with the innermost one.

    It is important when opening one of the brackets don't touch the rest of the expression, just rewriting it as is.
    Let's take the task above as an example.

    Example. Open the brackets and give like terms \(7x+2(5-(3x+y))\).
    Decision:


    Example. Expand the brackets and give like terms \(-(x+3(2x-1+(x-5)))\).
    Decision :

    \(-(x+3(2x-1\)\(+(x-5)\) \())\)

    This is a triple nesting of parentheses. We start with the innermost one (highlighted in green). There is a plus in front of the parenthesis, so it is simply removed.

    \(-(x+3(2x-1\)\(+x-5\) \())\)

    Now you need to open the second bracket, intermediate. But before that, we will simplify the expression by ghosting similar terms in this second bracket.

    \(=-(x\)\(+3(3x-6)\) \()=\)

    Now we open the second bracket (highlighted in blue). There is a multiplier in front of the parenthesis - so each term in the parenthesis is multiplied by it.

    \(=-(x\)\(+9x-18\) \()=\)

    And open the last parenthesis. Before the bracket minus - so all the signs are reversed.

    Bracket opening is a basic skill in mathematics. Without this skill, it is impossible to have a grade above three in grades 8 and 9. Therefore, I recommend a good understanding of this topic.

    In this article, we will consider in detail the basic rules for such an important topic in a mathematics course as opening brackets. You need to know the rules for opening brackets in order to correctly solve equations in which they are used.

    How to properly open parentheses when adding

    Expand the brackets preceded by the "+" sign

    This is the simplest case, because if there is an addition sign in front of the brackets, when the brackets are opened, the signs inside them do not change. Example:

    (9 + 3) + (1 - 6 + 9) = 9 + 3 + 1 - 6 + 9 = 16.

    How to open brackets preceded by a "-" sign

    In this case, you need to rewrite all the terms without brackets, but at the same time change all the signs inside them to the opposite ones. The signs change only for the terms from those brackets that were preceded by the “-” sign. Example:

    (9 + 3) - (1 - 6 + 9) = 9 + 3 - 1 + 6 - 9 = 8.

    How to open brackets when multiplying

    The parentheses are preceded by a multiplier

    In this case, you need to multiply each term by a factor and open the brackets without changing signs. If the multiplier has the sign "-", then when multiplying, the signs of the terms are reversed. Example:

    3 * (1 - 6 + 9) = 3 * 1 - 3 * 6 + 3 * 9 = 3 - 18 + 27 = 12.

    How to open two brackets with a multiplication sign between them

    In this case, you need to multiply each term from the first brackets with each term from the second brackets and then add the results. Example:

    (9 + 3) * (1 - 6 + 9) = 9 * 1 + 9 * (- 6) + 9 * 9 + 3 * 1 + 3 * (- 6) + 3 * 9 = 9 - 54 + 81 + 3 - 18 + 27 = 48.

    How to open brackets in a square

    If the sum or difference of two terms is squared, the brackets should be expanded according to the following formula:

    (x + y)^2 = x^2 + 2*x*y + y^2.

    In the case of a minus inside the brackets, the formula does not change. Example:

    (9 + 3) ^ 2 = 9 ^ 2 + 2 * 9 * 3 + 3 ^ 2 = 144.

    How to open parentheses in a different degree

    If the sum or difference of the terms is raised, for example, to the 3rd or 4th power, then you just need to break the degree of the bracket into “squares”. The powers of the same factors are added, and when dividing, the degree of the divisor is subtracted from the degree of the dividend. Example:

    (9 + 3) ^ 3 = ((9 + 3) ^ 2) * (9 + 3) = (9 ^ 2 + 2 * 9 * 3 + 3 ^ 2) * 12 = 1728.

    How to open 3 brackets

    There are equations in which 3 brackets are multiplied at once. In this case, you must first multiply the terms of the first two brackets among themselves, and then multiply the sum of this multiplication by the terms of the third bracket. Example:

    (1 + 2) * (3 + 4) * (5 - 6) = (3 + 4 + 6 + 8) * (5 - 6) = - 21.

    These bracket opening rules apply equally to both linear and trigonometric equations.

    Bracket expansion is a type of expression transformation. In this section, we will describe the rules for expanding brackets, as well as consider the most common examples of problems.

    Yandex.RTB R-A-339285-1

    What is parenthesis expansion?

    Parentheses are used to indicate the order in which actions are performed in numeric and alphabetic expressions, as well as in expressions with variables. It is convenient to pass from an expression with brackets to an identically equal expression without brackets. For example, replace the expression 2 (3 + 4) with an expression like 2 3 + 2 4 without brackets. This technique is called parenthesis opening.

    Definition 1

    Under the opening of brackets, we mean the methods of getting rid of brackets and are usually considered in relation to expressions that may contain:

    • signs "+" or "-" in front of brackets that contain sums or differences;
    • the product of a number, letter, or several letters, and the sum or difference, which is placed in brackets.

    This is how we used to consider the process of expanding brackets in the course school curriculum. However, no one prevents us from looking at this action more broadly. We can call parenthesis expansion the transition from an expression that contains negative numbers in parentheses to an expression that does not have parentheses. For example, we can go from 5 + (− 3) − (− 7) to 5 − 3 + 7 . In fact, this is also parenthesis expansion.

    In the same way, we can replace the product of expressions in brackets of the form (a + b) · (c + d) with the sum a · c + a · d + b · c + b · d . This technique also does not contradict the meaning of parentheses expansion.

    Here is another example. We can assume that in expressions, instead of numbers and variables, any expressions can be used. For example, the expression x 2 1 a - x + sin (b) will correspond to an expression without brackets of the form x 2 1 a - x 2 x + x 2 sin (b) .

    One more point deserves special attention, which concerns the peculiarities of writing solutions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as equality. For example, after opening the parentheses, instead of the expression 3 − (5 − 7) we get the expression 3 − 5 + 7 . We can write both of these expressions as the equality 3 − (5 − 7) = 3 − 5 + 7 .

    Carrying out actions with cumbersome expressions may require recording intermediate results. Then the solution will have the form of a chain of equalities. For example, 5 − (3 − (2 − 1)) = 5 − (3 − 2 + 1) = 5 − 3 + 2 − 1 or 5 − (3 − (2 − 1)) = 5 − 3 + (2 − 1) = 5 − 3 + 2 − 1 .

    Rules for opening brackets, examples

    Let's start with the rules for opening parentheses.

    Single numbers in brackets

    Negative numbers in parentheses often appear in expressions. For example, (− 4) and 3 + (− 4) . Positive numbers in brackets also take place.

    Let us formulate the rule for opening brackets that contain single positive numbers. Suppose a is any positive number. Then we can replace (a) with a, + (a) with + a, - (a) with - a. If instead of a we take a specific number, then according to the rule: the number (5) will be written as 5 , the expression 3 + (5) without brackets will take the form 3 + 5 , since + (5) is replaced by + 5 , and the expression 3 + (− 5) is equivalent to the expression 3 − 5 , as + (− 5) is replaced by − 5 .

    Positive numbers are usually written without using parentheses, since the parentheses are redundant in this case.

    Now consider the rule for opening brackets that contain a single negative number. + (−a) we replace with − a, − (− a) is replaced by + a . If the expression starts with a negative number (-a), which is written in brackets, then the brackets are omitted and instead of (-a) remains − a.

    Here are some examples: (− 5) can be written as − 5 , (− 3) + 0 , 5 becomes − 3 + 0 , 5 , 4 + (− 3) becomes 4 − 3 , and − (− 4) − (− 3) after opening the brackets takes the form 4 + 3 , since − (− 4) and − (− 3) is replaced by + 4 and + 3 .

    It should be understood that the expression 3 · (− 5) cannot be written as 3 · − 5. This will be discussed in the following paragraphs.

    Let's see what the parenthesis expansion rules are based on.

    According to the rule, the difference a − b is equal to a + (− b) . Based on the properties of actions with numbers, we can make a chain of equalities (a + (− b)) + b = a + ((− b) + b) = a + 0 = a which will be fair. This chain of equalities, by virtue of the meaning of subtraction, proves that the expression a + (− b) is the difference a-b.

    Based on the properties of opposite numbers and the rules for subtracting negative numbers, we can assert that − (− a) = a , a − (− b) = a + b .

    There are expressions that are made up of a number, minus signs and several pairs of brackets. Using the above rules allows you to sequentially get rid of brackets, moving from inner brackets to outer ones or vice versa. An example of such an expression would be − (− ((− (5)))) . Let's open the brackets, moving from the inside to the outside: − (− ((− (5)))) = − (− ((− 5))) = − (− (− 5)) = − (5) = − 5 . This example can also be parsed in reverse: − (− ((− (5)))) = ((− (5))) = (− (5)) = − (5) = − 5 .

    Under a and b can be understood not only as numbers, but also as arbitrary numerical or literal expressions with a "+" in front that are not sums or differences. In all these cases, you can apply the rules in the same way as we did with single numbers in brackets.

    For example, after opening the brackets, the expression − (− 2 x) − (x 2) + (− 1 x) − (2 x y 2: z) takes the form 2 x − x 2 − 1 x − 2 x y 2: z . How did we do it? We know that − (− 2 x) is + 2 x , and since this expression comes first, then + 2 x can be written as 2 x , - (x 2) = - x 2, + (− 1 x) = − 1 x and − (2 x y 2: z) = − 2 x y 2: z.

    In the products of two numbers

    Let's start with the rule for expanding brackets in the product of two numbers.

    Let's pretend that a and b are two positive numbers. In this case, the product of two negative numbers − a and − b of the form (− a) (− b) can be replaced by (a b) , and the products of two numbers with opposite signs of the form (− a) b and a (− b) can be replaced by (− a b). Multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus, gives a minus.

    The correctness of the first part of the written rule is confirmed by the rule for multiplying negative numbers. To confirm the second part of the rule, we can use the rules for multiplying numbers with different signs.

    Let's look at a few examples.

    Example 1

    Consider the algorithm for opening brackets in the product of two negative numbers - 4 3 5 and - 2 , of the form (- 2) · - 4 3 5 . To do this, we replace the original expression with 2 · 4 3 5 . Let's expand the brackets and get 2 · 4 3 5 .

    And if we take the quotient of negative numbers (− 4) : (− 2) , then the record after opening the brackets will look like 4: 2

    Instead of negative numbers − a and − b can be any expressions with a leading minus sign that are not sums or differences. For example, these can be products, partials, fractions, degrees, roots, logarithms, trigonometric functions etc.

    Let's open the brackets in the expression - 3 · x x 2 + 1 · x · (- ln 5) . According to the rule, we can make the following transformations: - 3 x x 2 + 1 x (- ln 5) = - 3 x x 2 + 1 x ln 5 = 3 x x 2 + 1 x ln 5 .

    Expression (− 3) 2 can be converted to the expression (− 3 2) . After that, you can open the brackets: − 3 2.

    2 3 - 4 5 = - 2 3 4 5 = - 2 3 4 5

    Dividing numbers with different signs may also require the preliminary expansion of brackets: (− 5) : 2 = (− 5: 2) = − 5: 2 and 2 3 4: (- 3 , 5) = - 2 3 4: 3 , 5 = - 2 3 4: 3 , 5 .

    The rule can be used to perform multiplication and division of expressions with different signs. Let's give two examples.

    1 x + 1: x - 3 = - 1 x + 1: x - 3 = - 1 x + 1: x - 3

    sin (x) (- x 2) \u003d (- sin (x) x 2) \u003d - sin (x) x 2

    In the products of three or more numbers

    Let's move on to the product and the quotients, which contain large quantity numbers. For expanding parentheses, here will act next rule. At even number negative numbers, you can omit the brackets, replacing the numbers with their opposites. After that, you need to enclose the resulting expression in new brackets. For an odd number of negative numbers, omitting the brackets, replace the numbers with their opposites. After that, the resulting expression must be taken in new brackets and put a minus sign in front of it.

    Example 2

    For example, let's take the expression 5 · (− 3) · (− 2) , which is the product of three numbers. There are two negative numbers, so we can write the expression as (5 3 2) and then finally open the brackets, getting the expression 5 3 2 .

    In the product (− 2 , 5) (− 3) : (− 2) 4: (− 1 , 25) : (− 1) five numbers are negative. so (− 2 , 5) (− 3) : (− 2) 4: (− 1 , 25) : (− 1) = (− 2 . 5 3: 2 4: 1 , 25: 1) . Finally opening the brackets, we get −2.5 3:2 4:1.25:1.

    The above rule can be justified as follows. First, we can rewrite such expressions as a product, replacing division with multiplication by the reciprocal. We represent each negative number as the product of a multiplier and replace - 1 or - 1 with (− 1) a.

    Using the commutative property of multiplication, we swap the factors and transfer all factors equal to − 1 , to the beginning of the expression. The product of an even number minus ones is equal to 1, and an odd number is equal to − 1 , which allows us to use the minus sign.

    If we did not use the rule, then the chain of actions for opening brackets in the expression - 2 3: (- 2) 4: - 6 7 would look like this:

    2 3: (- 2) 4: - 6 7 = - 2 3 - 1 2 4 - 7 6 = = (- 1) 2 3 (- 1) 1 2 4 (- 1 ) 7 6 = = (- 1) (- 1) (- 1) 2 3 1 2 4 7 6 = (- 1) 2 3 1 2 4 7 6 = = - 2 3 1 2 4 7 6

    The above rule can be used when expanding brackets in expressions that are products and quotients with a minus sign that are not sums or differences. Take for example the expression

    x 2 (- x) : (- 1 x) x - 3: 2 .

    It can be reduced to an expression without brackets x 2 · x: 1 x · x - 3: 2 .

    Opening parentheses preceded by a + sign

    Consider a rule that can be applied to expand brackets that are preceded by a plus sign and the "contents" of those brackets are not multiplied or divided by any number or expression.

    According to the rule, brackets together with the sign in front of them are omitted, while the signs of all terms in brackets are preserved. If there is no sign in front of the first term in brackets, then you need to put a plus sign.

    Example 3

    For example, we give the expression (12 − 3 , 5) − 7 . By omitting the brackets, we keep the signs of the terms in the brackets and put a plus sign in front of the first term. The entry will look like (12 − ​​3 , 5) − 7 = + 12 − 3 , 5 − 7 . In the above example, it is not necessary to put a sign in front of the first term, since + 12 - 3, 5 - 7 = 12 - 3, 5 - 7.

    Example 4

    Let's consider one more example. Take the expression x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x and perform actions with it x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x = = x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x

    Here is another example of expanding parentheses:

    Example 5

    2 + x 2 + 1 x - x y z + 2 x - 1 + (- 1 + x - x 2) = = 2 + x 2 + 1 x - x y z + 2 x - 1 - 1 + x + x2

    How to expand parentheses preceded by a minus sign

    Consider cases where there is a minus sign in front of the brackets, and which are not multiplied (or divided) by any number or expression. According to the rule for expanding brackets preceded by the “-” sign, the brackets with the “-” sign are omitted, while the signs of all terms inside the brackets are reversed.

    Example 6

    For example:

    1 2 \u003d 1 2, - 1 x + 1 \u003d - 1 x + 1, - (- x 2) \u003d x 2

    Variable expressions can be converted using the same rule:

    X + x 3 - 3 - - 2 x 2 + 3 x 3 x + 1 x - 1 - x + 2,

    we get x - x 3 - 3 + 2 x 2 - 3 x 3 x + 1 x - 1 - x + 2 .

    Opening parentheses when multiplying a number by a parenthesis, expressions by a parenthesis

    Here we will consider cases when it is necessary to open brackets that are multiplied or divided by any number or expression. Here formulas of the form (a 1 ± a 2 ± ... ± a n) b = (a 1 b ± a 2 b ± ... ± a n b) or b (a 1 ± a 2 ± … ± a n) = (b a 1 ± b a 2 ± … ± b a n), where a 1 , a 2 , … , a n and b are some numbers or expressions.

    Example 7

    For example, let's expand the brackets in the expression (3 − 7) 2. According to the rule, we can make the following transformations: (3 − 7) 2 = (3 2 − 7 2) . We get 3 · 2 − 7 · 2 .

    Expanding the brackets in the expression 3 x 2 1 - x + 1 x + 2, we get 3 x 2 1 - 3 x 2 x + 3 x 2 1 x + 2.

    Multiply a parenthesis by a parenthesis

    Consider the product of two brackets of the form (a 1 + a 2) · (b 1 + b 2) . This will help us get a rule for expanding parentheses when multiplying a parenthesis by a parenthesis.

    In order to solve the above example, we denote the expression (b 1 + b 2) like b. This will allow us to use the parenthesis-expression multiplication rule. We get (a 1 + a 2) (b 1 + b 2) = (a 1 + a 2) b = (a 1 b + a 2 b) = a 1 b + a 2 b . By doing a reverse substitution b on (b 1 + b 2), again apply the rule for multiplying the expression by the bracket: a 1 b + a 2 b = = a 1 (b 1 + b 2) + a 2 (b 1 + b 2) = = (a 1 b 1 + a 1 b 2) + (a 2 b 1 + a 2 b 2) = = a 1 b 1 + a 1 b 2 + a 2 b 1 + a 2 b 2

    Thanks to a number of simple tricks, we can come to the sum of the products of each of the terms from the first bracket and each of the terms from the second bracket. The rule can be extended to any number of terms inside the brackets.

    Let us formulate the rules for multiplying brackets by brackets: in order to multiply two sums among themselves, it is necessary to multiply each of the terms of the first sum by each of the terms of the second sum and add the results.

    The formula will look like:

    (a 1 + a 2 + . . . + a m) (b 1 + b 2 + . . . + b n) = = a 1 b 1 + a 1 b 2 + . . . + a 1 b n + + a 2 b 1 + a 2 b 2 + . . . + a 2 b n + + . . . + + a m b 1 + a m b 1 + . . . a m b n

    Let's expand the brackets in the expression (1 + x) · (x 2 + x + 6) It is a product of two sums. Let's write the solution: (1 + x) (x 2 + x + 6) = = (1 x 2 + 1 x + 1 6 + x x 2 + x x + x 6) = = 1 x 2 + 1 x + 1 6 + x x 2 + x x + x 6

    Separately, it is worth dwelling on those cases when there is a minus sign in brackets along with plus signs. For example, let's take the expression (1 − x) · (3 · x · y − 2 · x · y 3) .

    First, we represent the expressions in brackets as sums: (1 + (− x)) (3 x y + (− 2 x y 3)). Now we can apply the rule: (1 + (− x)) (3 x y + (− 2 x y 3)) = = (1 3 x y + 1 (− 2 x y 3) + (− x) 3 x y + (− x) (− 2 x y 3))

    Let's expand the brackets: 1 3 x y − 1 2 x y 3 − x 3 x y + x 2 x y 3 .

    Parentheses expansion in products of several brackets and expressions

    If there are three or more expressions in brackets in the expression, it is necessary to expand the brackets sequentially. It is necessary to start the transformation with the fact that the first two factors are taken in brackets. Inside these brackets, we can perform transformations according to the rules discussed above. For example, the parentheses in the expression (2 + 4) 3 (5 + 7 8) .

    The expression contains three factors at once (2 + 4) , 3 and (5 + 7 8) . We will expand the brackets sequentially. We enclose the first two factors in one more brackets, which we will make red for clarity: (2 + 4) 3 (5 + 7 8) = ((2 + 4) 3) (5 + 7 8).

    In accordance with the rule of multiplying a bracket by a number, we can carry out the following actions: ((2 + 4) 3) (5 + 7 8) = (2 3 + 4 3) (5 + 7 8) .

    Multiply bracket by bracket: (2 3 + 4 3) (5 + 7 8) = 2 3 5 + 2 3 7 8 + 4 3 5 + 4 3 7 8 .

    Parenthesis in kind

    Degrees, the bases of which are some expressions written in brackets, with natural exponents can be considered as a product of several brackets. Moreover, according to the rules from the two previous paragraphs, they can be written without these brackets.

    Consider the process of transforming the expression (a + b + c) 2 . It can be written as a product of two brackets (a + b + c) (a + b + c). We multiply bracket by bracket and get a a + a b + a c + b a + b b + b c + c a + c b + c c .

    Let's take another example:

    Example 8

    1 x + 2 3 = 1 x + 2 1 x + 2 1 x + 2 = = 1 x 1 x + 1 x 2 + 2 1 x + 2 2 1 x + 2 = = 1 x 1 x 1 x + 1 x 2 1 x + 2 1 x 1 x + 2 2 1 x + 1 x 1 x 2 + + 1 x 2 2 + 2 1 x 2 + 2 2 2

    Dividing a parenthesis by a number and a parenthesis by a parenthesis

    Dividing a parenthesis by a number suggests that you must divide by the number all the terms enclosed in brackets. For example, (x 2 - x) : 4 = x 2: 4 - x: 4 .

    Division can be previously replaced by multiplication, after which you can use suitable rule opening brackets in a work. The same rule applies when dividing a parenthesis by a parenthesis.

    For example, we need to open the brackets in the expression (x + 2) : 2 3 . To do this, first replace the division by multiplying by the reciprocal of (x + 2) : 2 3 = (x + 2) · 2 3 . Multiply the bracket by the number (x + 2) 2 3 = x 2 3 + 2 2 3 .

    Here is another example of parenthesis division:

    Example 9

    1 x + x + 1: (x + 2) .

    Let's replace division with multiplication: 1 x + x + 1 1 x + 2 .

    Let's do the multiplication: 1 x + x + 1 1 x + 2 = 1 x 1 x + 2 + x 1 x + 2 + 1 1 x + 2 .

    Bracket expansion order

    Now consider the order of application of the rules discussed above in the expressions general view, i.e. in expressions that contain sums with differences, products with quotients, brackets in kind.

    The order of actions:

    • the first step is to raise the parentheses to a natural power;
    • at the second stage, brackets are opened in works and private;
    • the final step is to open the brackets in the sums and differences.

    Let's consider the order of actions using the example of the expression (− 5) + 3 · (− 2) : (− 4) − 6 · (− 7) . Let us transform from the expressions 3 (− 2) : (− 4) and 6 (− 7) , which should take the form (3 2:4) and (− 6 7) . Substituting the obtained results into the original expression, we obtain: (− 5) + 3 (− 2) : (− 4) − 6 (− 7) = (− 5) + (3 2: 4) − (− 6 7). Expand the brackets: − 5 + 3 2: 4 + 6 7 .

    When dealing with expressions that contain parentheses within parentheses, it is convenient to perform transformations from the inside out.

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