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.

Now we will just move on to opening brackets in expressions in which the expression in brackets is multiplied by a number or expression. Let us formulate the rule for opening brackets preceded by a minus sign: the brackets together with the minus sign are omitted, and the signs of all terms in brackets are replaced by opposite ones.

One type of expression transformation is parentheses expansion. Numeric, literal and variable expressions are composed using brackets, which can indicate the order in which actions are performed, contain a negative number, etc. Let's assume that in the expressions described above, instead of numbers and variables, there can be any expressions.

And let's pay attention to one more point concerning the peculiarities of writing the solution when opening the brackets. In the previous paragraph, we dealt with what is called parenthesis expansion. To do this, there are rules for opening brackets, which we now review. This rule is dictated by the fact that it is customary to write positive numbers without brackets, brackets in this case are unnecessary. The expression (−3.7)−(−2)+4+(−9) can be written without brackets as −3.7+2+4−9.

Finally, the third part of the rule is simply due to the peculiarities of writing negative numbers on the left in the expression (which we mentioned in the brackets section for writing negative numbers). You may encounter expressions made up of a number, minus signs, and multiple pairs of parentheses. If you expand the brackets, moving from inner to outer, then the solution will be: −(−((−(5))))=−(−((−5)))=−(−(−5))=−( 5)=−5.

How to open brackets?

Here is an explanation: −(−2 x) is +2 x, and since this expression comes first, then +2 x can be written as 2 x, −(x2)=−x2, +(−1/ x)=−1/x and −(2 x y2:z)=−2 x y2:z. The first part of the written rule for opening brackets follows directly from the rule for multiplying negative numbers. The second part of it is a consequence of the rule for multiplying numbers with different signs. Let's move on to examples of expanding brackets in products and quotients of two numbers with different signs.

Bracket opening: rules, examples, solutions.

The above rule takes into account the entire chain of these actions and significantly speeds up the process of opening brackets. The same rule allows you to open brackets in expressions that are products and partial expressions with a minus sign that are not sums and differences.

Consider examples of the application of this rule. We give the corresponding rule. Above, we have already encountered expressions of the form −(a) and −(−a), which without brackets are written as −a and a, respectively. For example, −(3)=3, and. These are special cases of the stated rule. Now consider examples of opening brackets when sums or differences are enclosed in them. We will show examples of the use of this rule. Denote the expression (b1+b2) as b, after which we use the rule for multiplying the bracket by the expression from the previous paragraph, we have (a1+a2) (b1+b2)=(a1+a2) b=(a1 b+a2 b)=a1 b+a2 b.

By induction, this statement can be extended to an arbitrary number of terms in each bracket. It remains to open the brackets in the resulting expression, using the rules from the previous paragraphs, as a result, we get 1 3 x y−1 2 x y3−x 3 x y+x 2 x y3.

The rule in mathematics is the opening of brackets if there is (+) and (-) in front of the brackets, a very necessary rule

This expression is the product of three factors (2+4), 3 and (5+7 8). The brackets must be opened sequentially. Now we use the rule for multiplying a bracket by a number, we have ((2+4) 3) (5+7 8)=(2 3+4 3) (5+7 8). Degrees, the bases of which are some expressions written in brackets, with natural exponents can be considered as a product of several brackets.

For example, let's transform the expression (a+b+c)2. First, we write it as a product of two brackets (a + b + c) (a + b + c), now we multiply the bracket by bracket, we get a a + a b + a c + b a + b b+b c+c a+c b+c c.

We also say that to raise the sums and differences of two numbers to a natural power, it is advisable to use the Newton binomial formula. For example, (5+7−3):2=5:2+7:2−3:2. It is no less convenient to preliminarily replace division with multiplication, and then use the appropriate rule for opening brackets in the product.

It remains to figure out the order of opening brackets using examples. Take the expression (−5)+3 (−2):(−4)−6 (−7). Substitute these results in the original expression: (−5)+3 (−2):(−4)−6 (−7)=(−5)+(3 2:4)−(−6 7) . It remains only to complete the opening of the brackets, as a result we have −5+3 2:4+6 7. This means that when passing from the left side of the equality to the right side, the brackets were opened.

Note that in all three examples, we simply removed the parentheses. First, add 445 to 889. This mental action can be performed, but it is not very easy. Let's open the brackets and see that the changed order of operations will greatly simplify the calculations.

How to open parentheses in a different degree

Illustrative example and rule. Consider an example: . You can find the value of the expression by adding 2 and 5, and then taking the resulting number with the opposite sign. The rule does not change if there are not two, but three or more terms in brackets. Comment. Signs are reversed only in front of the terms. In order to open the brackets, in this case, we need to recall the distributive property.

Single numbers in brackets

Your mistake is not in the signs, but in the wrong work with fractions? In 6th grade we got acquainted with positive and negative numbers. How will we solve examples and equations?

How much is in brackets? What can be said about these expressions? Of course, the result of the first and second examples is the same, so you can put an equal sign between them: -7 + (3 + 4) = -7 + 3 + 4. So what did we do with the brackets?

Demonstration of slide 6 with the rules for opening brackets. Thus, the rules for opening brackets will help us solve examples, simplify expressions. Next, students are invited to work in pairs: it is necessary to connect the expression containing brackets with the corresponding expression without brackets with arrows.

Slide 11 Once in the Sunny City, Znayka and Dunno argued which of them solved the equation correctly. Next, students independently solve the equation, applying the rules for opening brackets. Solving equations ”Lesson objectives: educational (fixing ZUNs on the topic:“ Opening brackets.

Lesson topic: “Opening parentheses. 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. First, the first two factors are taken, enclosed in one more brackets, and inside these brackets, the brackets are opened according to one of the already known rules.

rawalan.freezeet.ru

Bracket opening: rules and examples (Grade 7)

The main function of brackets is to change the order of actions when calculating values numeric expressions . 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: \(5·(3+7)=5·10=50\).

However, if we are dealing with algebraic expression containing variable- for example, like this: \ (2 (x-3) \) - then it is impossible to calculate the value in the bracket, the variable interferes. Therefore, in this case, the brackets are “opened”, using the appropriate rules for this.

Bracket expansion rules

If there is a plus sign before the bracket, then the bracket is simply removed, the expression in it remains unchanged. In other words:

Here it is necessary to clarify that in mathematics, to reduce entries, it is customary not to write the plus sign if it is the first in the expression. For example, if we add two positive numbers, for example, seven and three, then we do not write \(+7+3\), 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.

Example . Open the bracket and give like terms: \((x-11)+(2+3x)\).
Decision : \((x-11)+(2+3x)=x-11+2+3x=4x-9\).

If there is a minus sign in front of the bracket, then when the bracket is removed, each member of the expression inside it changes sign to the opposite:

Here it is necessary to clarify that a, while it was in brackets, had a plus sign (they just didn’t write it), and after removing the bracket, this plus changed to a minus.

Example : Simplify the expression \(2x-(-7+x)\).
Decision : there are two terms inside the bracket: \(-7\) and \(x\), and there is a minus before the bracket. This means that the signs will change - and the seven will now be with a plus, and the x with a minus. open the bracket and bring like terms .

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

If there is a factor in front of the bracket, then each member of the bracket is multiplied by it, that is:

Example. Expand the brackets \(5(3-x)\).
Decision : We have \(3\) and \(-x\) in the parenthesis, and a five in front of the parenthesis. 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 \(-2\).

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:

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. We 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...

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:

Let's start the task by opening the inner bracket (the one inside). Opening it, we are only dealing with the fact that it is directly related to it - this is the bracket itself and the minus in front of it (highlighted in green). Everything else (not selected) is rewritten as it was.

Solving problems in mathematics online

Online calculator.
Polynomial simplification.
Multiplication of polynomials.

With this math program, you can simplify a polynomial.
While the program is running:
- multiplies polynomials
- sums monomials (gives like ones)
- opens brackets
- Raises a polynomial to a power

The polynomial simplification program does not just give the answer to the problem, it gives a detailed solution with explanations, i.e. displays the solution process so that you can check your knowledge of mathematics and / or algebra.

This program can be useful for students of general education schools in preparing for tests and exams, when testing knowledge before the Unified State Examination, and for parents to control the solution of many problems in mathematics and algebra. Or maybe it's too expensive for you to hire a tutor or buy new textbooks? Or do you just want to get your math or algebra homework done as quickly as possible? In this case, you can also use our programs with a detailed solution.

In this way, you can conduct your own training and/or the training of your younger brothers or sisters, while the level of education in the field of tasks to be solved is increased.

Because There are a lot of people who want to solve the problem, your request is queued.
After a few seconds, the solution will appear below.
Please wait sec.

A bit of theory.

The product of a monomial and a polynomial. The concept of a polynomial

Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:

The sum of monomials is called a polynomial. The terms in a polynomial are called members of the polynomial. Mononomials are also referred to as polynomials, considering a monomial as a polynomial consisting of one member.

We represent all the terms as monomials of the standard form:

We give similar terms in the resulting polynomial:

The result is a polynomial, all members of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.

Behind polynomial degree standard form take the largest of the powers of its members. So, a binomial has a third degree, and a trinomial has a second.

Usually, the terms of standard form polynomials containing one variable are arranged in descending order of its exponents. For example:

The sum of several polynomials can be converted (simplified) into a standard form polynomial.

Sometimes the members of a polynomial need to be divided into groups, enclosing each group in parentheses. Since parentheses are the opposite of parentheses, it is easy to formulate parentheses opening rules:

If the + sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.

If a "-" sign is placed in front of the brackets, then the terms enclosed in brackets are written with opposite signs.

Transformation (simplification) of the product of a monomial and a polynomial

Using the distributive property of multiplication, one can transform (simplify) the product of a monomial and a polynomial into a polynomial. For example:

The product of a monomial and a polynomial is identically equal to the sum of the products of this monomial and each of the terms of the polynomial.

This result is usually formulated as a rule.

To multiply a monomial by a polynomial, one must multiply this monomial by each of the terms of the polynomial.

We have repeatedly used this rule for multiplying by a sum.

The product of polynomials. Transformation (simplification) of the product of two polynomials

In general, the product of two polynomials is identically equal to the sum of the product of each term of one polynomial and each term of the other.

Usually use the following rule.

To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other and add the resulting products.

Abbreviated multiplication formulas. Sum, Difference, and Difference Squares

Some expressions in algebraic transformations have to be dealt with more often than others. Perhaps the most common expressions are and, that is, the square of the sum, the square of the difference, and the difference of squares. You have noticed that the names of these expressions seem to be incomplete, so, for example, - this, of course, is not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b is not so common, as a rule, instead of the letters a and b, it contains various, sometimes quite complex expressions.

Expressions are easy to convert (simplify) into polynomials of the standard form, in fact, you have already met with such a task when multiplying polynomials:

The resulting identities are useful to remember and apply without intermediate calculations. Short verbal formulations help this.

- the square of the sum is equal to the sum of squares and twice the product.

- the square of the difference is equal to the sum of the squares without the double product.

- the difference of squares is equal to the product of the difference by the sum.

These three identities allow in transformations to replace their left parts with right ones and vice versa - right parts with left ones. The most difficult thing in this case is to see the corresponding expressions and understand what the variables a and b are replaced in them. Let's look at a few examples of using abbreviated multiplication formulas.

Books (textbooks) Abstracts of the Unified State Examination and OGE tests online Games, puzzles Graphing of functions Spelling Dictionary of the Russian Language Dictionary of youth slang Directory of Russian schools Catalog of secondary schools in Russia Catalog of Russian universities numeric fractions Solving problems for percentages Complex numbers: sum, difference, product and quotient Systems of 2 linear equations with two variables Solving a quadratic equation Sorting out the square of a binomial and factoring a square trinomial Solving inequalities Solving systems of inequalities Building a graph of a quadratic function Building a graph of a fractional linear function Solving arithmetic and geometric progressions Solving trigonometric, exponential, logarithmic equations Calculating limits, derivatives, tangents Integral, antiderivative Solving triangles Calculating actions with vectors Calculating actions actions with lines and planes Area of ​​geometric shapes Perimeter of geometric shapes Volume of geometric bodies Surface area of ​​geometric bodies
Traffic situations constructor
Weather - news - horoscopes

www.mathsolution.ru

Bracket expansion

We continue to study the basics of algebra. In this lesson, we will learn how to open parentheses in expressions. To expand brackets means to rid the expression of these brackets.

To open brackets, you need to learn by heart only two rules. With regular practice, you can open the brackets with your eyes closed, and those rules that needed to be memorized by heart can be safely forgotten.

The first rule of parenthesis expansion

Consider the following expression:

The value of this expression is 2 . Let's open the brackets in this expression. To expand parentheses means to get rid of them without affecting the meaning of the expression. That is, after getting rid of the brackets, the value of the expression 8+(−9+3) should still be equal to two.

The first parenthesis expansion rule looks like this:

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

So we see that in the expression 8+(−9+3) there is a plus in front of the brackets. This plus must be omitted along with the parentheses. In other words, the brackets will disappear along with the plus that stood in front of them. And what was in brackets will be written unchanged:

8−9+3 . This expression is equal to 2 , like the previous parenthesized expression was equal to 2 .

8+(−9+3) and 8−9+3

8 + (−9 + 3) = 8 − 9 + 3

Example 2 Expand brackets in an expression 3 + (−1 − 4)

There is a plus in front of the brackets, so this plus is omitted along with the brackets. What was in the brackets will remain unchanged:

3 + (−1 − 4) = 3 − 1 − 4

Example 3 Expand brackets in an expression 2 + (−1)

In this example, the expansion of brackets has become a kind of inverse operation of replacing subtraction with addition. What does it mean?

In the expression 2−1 subtraction occurs, but it can be replaced by addition. Then you get the expression 2+(−1) . But if in the expression 2+(−1) open the brackets, you get the original 2−1 .

Therefore, the first bracket expansion rule can be used to simplify expressions after some transformations. That is, rid it of brackets and make it easier.

For example, let's simplify the expression 2a+a−5b+b .

To simplify this expression, we can add like terms. Recall that to reduce like terms, you need to add the coefficients of like terms and multiply the result by the common letter part:

Got an expression 3a+(−4b). In this expression, open the brackets. There is a plus before the brackets, so we use the first rule for opening brackets, that is, we omit the brackets along with the plus that comes before these brackets:

So the expression 2a+a−5b+b simplified to 3a−4b .

Having opened one brackets, others may meet along the way. We apply the same rules to them as to the first. For example, let's expand the brackets in the following expression:

There are two places where you need to expand the brackets. In this case, the first rule for expanding parentheses applies, namely, omitting the parentheses along with the plus that comes before these parentheses:

2 + (−3 + 1) + 3 + (−6) = 2 − 3 + 1 + 3 − 6

Example 3 Expand brackets in an expression 6+(−3)+(−2)

In both places where there are brackets, they are preceded by a plus sign. Here again, the first parenthesis expansion rule applies:

Sometimes the first term in brackets is written without a sign. For example, in the expression 1+(2+3−4) first term in brackets 2 written without a sign. The question arises, what sign will come before the deuce after the brackets and the plus in front of the brackets are omitted? The answer suggests itself - there will be a plus in front of the deuce.

In fact, even being in brackets, there is a plus in front of the deuce, but we do not see it due to the fact that it is not written down. We have already said that the full notation of positive numbers looks like +1, +2, +3. But the pluses are not traditionally written down, which is why we see the positive numbers that are familiar to us. 1, 2, 3 .

Therefore, to open parentheses in an expression 1+(2+3−4) , you need to omit the brackets as usual along with the plus in front of these brackets, but write the first term that was in brackets with a plus sign:

1 + (2 + 3 − 4) = 1 + 2 + 3 − 4

Example 4 Expand brackets in an expression −5 + (2 − 3)

There is a plus in front of the brackets, so we apply the first rule for opening brackets, namely, we omit the brackets along with the plus that comes before these brackets. But the first term, which is written in brackets with a plus sign:

−5 + (2 − 3) = −5 + 2 − 3

Example 5 Expand brackets in an expression (−5)

There is a plus before the parenthesis, but it is not written due to the fact that there were no other numbers or expressions before it. Our task is to remove the brackets by applying the first rule for expanding brackets, namely, omitting the brackets along with this plus (even if it is invisible)

Example 6 Expand brackets in an expression 2a + (−6a + b)

There is a plus in front of the brackets, so this plus is omitted along with the brackets. What was in brackets will be written unchanged:

2a + (−6a + b) = 2a −6a + b

Example 7 Expand brackets in an expression 5a + (−7b + 6c) + 3a + (−2d)

In this expression, there are two places where you need to open the brackets. In both sections, there is a plus in front of the brackets, which means that this plus is omitted along with the brackets. What was in brackets will be written unchanged:

5a + (−7b + 6c) + 3a + (−2d) = 5a −7b + 6c + 3a − 2d

The second rule for opening parentheses

Now let's look at the second parenthesis expansion rule. It is used when there is a minus before the parentheses.

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.

For example, let's expand the brackets in the following expression

We see that there is a minus before the brackets. So you need to apply the second expansion rule, namely, omit the brackets along with the minus in front of these brackets. In this case, the terms that were in brackets will change their sign to the opposite:

We got an expression without brackets 5+2+3 . This expression is equal to 10, just like the previous expression with brackets was equal to 10.

Thus, between expressions 5−(−2−3) and 5+2+3 you can put an equal sign, since they are equal to the same value:

5 − (−2 − 3) = 5 + 2 + 3

Example 2 Expand brackets in an expression 6 − (−2 − 5)

There is a minus before the brackets, so we apply the second rule for opening brackets, namely, we omit the brackets along with the minus that comes before these brackets. In this case, the terms that were in brackets are written with opposite signs:

6 − (−2 − 5) = 6 + 2 + 5

Example 3 Expand brackets in an expression 2 − (7 + 3)

There is a minus before the brackets, so we apply the second rule for opening brackets:

Example 4 Expand brackets in an expression −(−3 + 4)

Example 5 Expand brackets in an expression −(−8 − 2) + 16 + (−9 − 2)

There are two places where you need to expand the brackets. In the first case, you need to apply the second rule for opening brackets, and when the turn comes to the expression +(−9−2) you need to apply the first rule:

−(−8 − 2) + 16 + (−9 − 2) = 8 + 2 + 16 − 9 − 2

Example 6 Expand brackets in an expression −(−a−1)

Example 7 Expand brackets in an expression −(4a + 3)

Example 8 Expand brackets in an expression a −(4b + 3) + 15

Example 9 Expand brackets in an expression 2a + (3b − b) − (3c + 5)

There are two places where you need to expand the brackets. In the first case, you need to apply the first rule for expanding brackets, and when the turn comes to the expression −(3c+5) you need to apply the second rule:

2a + (3b − b) − (3c + 5) = 2a + 3b − b − 3c − 5

Example 10 Expand brackets in an expression -a − (−4a) + (−6b) − (−8c + 15)

There are three places where you need to expand the brackets. First you need to apply the second rule for expanding brackets, then the first, and then again the second:

-a - (-4a) + (-6b) - (-8c + 15) = −a + 4a - 6b + 8c - 15

Parentheses expansion mechanism

The rules for opening brackets, which we have now considered, are based on the distributive law of multiplication:

Actually opening brackets call the procedure when the common factor is multiplied by each term in brackets. As a result of such multiplication, the brackets disappear. For example, let's expand the brackets in the expression 3×(4+5)

3 × (4 + 5) = 3 × 4 + 3 × 5

Therefore, if you need to multiply a number by an expression in brackets (or multiply an expression in brackets by a number), you need to say open the brackets.

But how is the distributive law of multiplication related to the rules for opening brackets that we considered earlier?

The fact is that before any brackets there is a common factor. In the example 3×(4+5) common factor is 3 . And in the example a(b+c) common factor is a variable a.

If there are no numbers or variables before the brackets, then the common factor is 1 or −1 , depending on which character comes before the brackets. If there is a plus in front of the brackets, then the common factor is 1 . If there is a minus before the brackets, then the common factor is −1 .

For example, let's expand the brackets in the expression −(3b−1). There is a minus before the brackets, so you need to use the second rule for opening brackets, that is, omit the brackets along with the minus before the brackets. And the expression that was in brackets, write with opposite signs:

We expanded the parentheses using the parenthesis expansion rule. But these same brackets can be opened using the distributive law of multiplication. To do this, we first write the common factor 1 in front of the brackets, which was not written down:

The minus that used to stand in front of the brackets referred to this unit. Now you can open the brackets by applying the distributive law of multiplication. For this, the common factor −1 you need to multiply by each term in brackets and add the results.

For convenience, we replace the difference in brackets with the sum:

−1 (3b −1) = −1 (3b + (−1)) = −1 × 3b + (−1) × (−1) = −3b + 1

Like last time, we got the expression −3b+1. Everyone will agree that this time more time was spent on solving such a simple example. Therefore, it is more reasonable to use the ready-made rules for opening brackets, which we considered in this lesson:

But it doesn't hurt to know how these rules work.

In this lesson, we learned another identical transformation. Together with opening the brackets, putting the general out of the brackets and bringing like terms, it is possible to slightly expand the range of tasks to be solved. For example:

Here you need to perform two actions - first open the brackets, and then bring like terms. So, in order:

1) Expand the brackets:

2) We give like terms:

In the resulting expression −10b+(−1) you can open the brackets:

Example 2 Open brackets and add like terms in the following expression:

1) Expand the brackets:

2) We present similar terms. This time, to save time and space, we will not write down how the coefficients are multiplied by the common letter part

Example 3 Simplify Expression 8m+3m and find its value at m=−4

1) Let's simplify the expression first. To simplify the expression 8m+3m, you can take out the common factor in it m for brackets:

2) Find the value of the expression m(8+3) at m=−4. For this, in the expression m(8+3) instead of a variable m substitute the number −4

m(8 + 3) = −4 (8 + 3) = −4 × 8 + (−4) × 3 = −32 + (−12) = −44

Everywhere. Everywhere and everywhere, wherever you look, there are such constructions:

These "constructions" in literate people cause an ambiguous reaction. At least like "is it really so - right?".
In general, personally, I can’t understand where the “fashion” of not closing external quotes came from. The first and only analogy that comes up in this regard is the analogy with brackets. Nobody doubts that two brackets in a row are normal. For example: "Pay for the entire circulation (200 pieces (of which 100 are defective))". But in the normality of setting two quotes in a row, someone doubted (I wonder who was the first?) ... And now everyone without exception began to produce constructions like LLC Firm Pupkov and Co. with a clear conscience.
But even if you have not seen the rule in your life, which will be discussed below, then the only logically justified option (using the brackets as an example) would be the following: Firm Pupkov and Co LLC.
So, the rule itself:
If at the beginning or at the end of a quotation (the same applies to direct speech) there are internal and external quotation marks, then they must differ from each other in a pattern (the so-called "herringbones" and "cutes"), and external quotation marks should not be omitted, for example: C The sides of the ship were radioed: "Leningrad has entered the tropics and is continuing on its course." About Zhukovsky, Belinsky writes: “Contemporaries of Zhukovsky’s youth looked at him mainly as an author of ballads, and in one of his messages Batyushkov called him a “ballade player.”
© Rules of Russian spelling and punctuation. - Tula: Autograph, 1995. - 192 p.
Accordingly ... if you do not have the opportunity to type in quotes, "Christmas trees", then what can you do, you will have to use such "" icons. However, the impossibility (or unwillingness) to use Russian quotes is by no means the reason why you can not close the outer quotes.

Thus, it seems that they figured out the incorrect design of Firm Pupkov and Co LLC. There are also constructions of the type LLC Firm Pupkov and Co.
From the rule, it is quite clear that such constructions are illiterate ... (Correct: LLC Firm Pupkov and Co.

However!
Milchin's Publisher's and Author's Handbook (2004 edition) states that two design options can be used in such cases. The use of "herringbones" and "paws" and (in the absence of technical means) the use of only "herringbones": two opening and one closing.
The directory is “fresh” and personally I immediately have 2 questions here. Firstly, with what joy you can still use one closing quote-herringbone (well, this is illogical, see above), and secondly, the phrase “in the absence of technical means” especially attracts attention. How is that, sorry? Here, open Notepad and type “only Christmas trees: two opening and one closing” there. There are no such characters on the keyboard. Printing a Christmas tree doesn't work... The combination Shift + 2 produces the sign " (which, as you know, is not even a quotation mark). Now open Microsoft Word and press Shift + 2 again. The program will correct " to " (or " ). Well, it turns out that the rule that existed for more than a dozen years was taken and rewritten under Microsoft Word? Like, since the Word from "Firm" Pupkov and Co "does" Firm "Pupkov and Co", then now let it be acceptable and correct ???
It seems so. And if so, then there is every reason to doubt the correctness of such an innovation.

Yes, and one more clarification ... about the very "lack of technical means." The fact is that on any Windows computer there are always "technical means" for entering both "Christmas trees" and "paws", so this new "rule" (for me it is in quotes) is wrong from the very beginning!

All special characters in a font can be easily typed by knowing the corresponding number of that character. It is enough to hold down Alt and type on the NumLock keyboard (NumLock is pressed, the indicator light is on) the corresponding symbol number:

„ Alt + 0132 (left foot)
“ Alt + 0147 (right foot)
« Alt + 0171 (left herringbone)
» Alt + 0187 (right herringbone)

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.

If you want to include information related to body text, but that information doesn't fit into the body of a sentence or paragraph, you need to put that information in parentheses. Putting it in parentheses reduces its importance so that it doesn't detract from the main point of the text.

• Example: J. R. R. Tolkien (author of The Lord of the Rings) and C. S. Lewis (author of The Chronicles of Narnia) were regular members of the literary discussion group known as the Inklings.

Notes in brackets. Often, when you write a numerical value in words, it is helpful to also write that value in numbers. You can specify a numerical form by putting it in parentheses.

• Example: She has to pay seven hundred dollars ($700) in rent by the end of this week.

Use of numbers or letters when listing. When you need to list a series of information within a paragraph or sentence, numbering each paragraph can make the list less confusing. You must put the numbers or letters used for each item in parentheses.

• Example: A company is looking for a job candidate who (1) is disciplined, (2) knows everything there is to know about the latest trends in photo editing and software improvements, and (3) has at least five years of professional experience in the field.
• Example: A company is looking for a job candidate who (A) is disciplined, (B) knows everything there is to know about the latest trends in photo editing and software improvements, and (C) has at least five years of professional experience in the field.

Plural designation. In text, you can refer to something in the singular while also referring to the plural. If it is known that the reader will benefit from knowing that you mean both the plural and the singular, you can indicate your intention by putting in parentheses immediately after the noun the appropriate plural ending for that noun, if the noun has such shape.

• Example: The organizers of the festival this year are hoping for a large number of spectators, so be sure to purchase additional ticket(s).

Abbreviations notation. When writing the name of an organization, product, or other entity that typically has a well-known abbreviation, you must enter the entity's full name the first time you mention it in text. If you are going to refer to an object later using a well-known abbreviation, you must specify that abbreviation in parentheses so that readers know what to look for later.

• Example: Animal Welfare League (PLL) staff and volunteers hope to reduce and eventually eliminate animal cruelty and mistreatment within the community.

Mention of significant dates. Although not always necessary, in certain contexts you may be required to provide the date of birth and/or date of death of the specific person you are referring to in the text. Such dates must be enclosed in brackets.

• Example: Jane Austen (1775-1817) is known for her literary works Pride and Prejudice and Sense and Sensibility.
• George Martin (b. 1948) is the man behind the hit series Game of Thrones.

Use of introductory quotes. In nonfiction, introductory citations should be included when you directly or indirectly cite another work. These citations contain bibliographic information and should be enclosed in brackets immediately after the borrowed information.

• Example: Research shows that there is a link between migraine and clinical depression (Smith, 2012).
• Example: Research shows that there is a link between migraine and clinical depression (Smith 32).
• For more information on the correct use of introductory quotations in the text, see "How to use quotations in the text correctly."