If there is a then sign before the brackets. The rule for opening parentheses during a product

Parentheses are used to indicate the order in which actions are performed in numeric, literal, and variable expressions. It is convenient to move from an expression with brackets to an identically equal expression without brackets. This technique is called opening brackets.

Expanding parentheses means removing the parentheses from an expression.

One more point deserves special attention, which concerns the peculiarities of recording solutions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as an equality. For example, after expanding 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 shorten notations, it is customary not to write the plus sign if it appears first in an expression or in parentheses. 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 positive number. Similarly, if you see, for example, the expression (5+x) - know that before the bracket there is a plus, which is not written, and before the five there is a plus +(+5+x).

The rule for opening parentheses during addition

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

Example. Open the parentheses in the expression 2 + (7 + 3) Before the brackets there is a plus, which means we do not change the signs in front of the numbers in the brackets.

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

Rule for opening parentheses 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. Expand the parentheses in the expression 2 − (7 + 3)

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

2 − (7 + 3) = 2 − (+ 7 + 3)

When opening the brackets, we remove from the example the minus that was in front of 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. In this case, 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, the parentheses in the products are expanded in accordance with the distributive property of multiplication.

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

When multiplying bracket by bracket, each term in the first bracket is multiplied with each term in the second bracket.

(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: c(a−b)=ca−cb. Why? Because if you substitute one instead of c, you 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.

Opening parentheses when dividing

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

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

How to expand nested parentheses

If an expression contains nested parentheses, they are expanded in order, starting with the outer or inner ones.

In this case, it is important that when opening one of the brackets, do not touch the remaining brackets, simply rewriting them as is.

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

Now we'll move on to opening parentheses in expressions in which the expression in parentheses is multiplied by a number or expression. Let us formulate a rule for opening parentheses preceded by a minus sign: the parentheses together with the minus sign are omitted, and the signs of all terms in the parentheses are replaced with their opposites.

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

And let us pay attention to one more point regarding the peculiarities of writing a solution when opening brackets. In the previous paragraph, we dealt with what is called opening parentheses. To do this, there are rules for opening brackets, which we will now review. This rule is dictated by the fact that positive numbers are usually written without parentheses; in this case, parentheses are unnecessary. The expression (−3.7)−(−2)+4+(−9) can be written without parentheses 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 section on brackets for writing negative numbers). You may encounter expressions made up of a number, minus signs, and several pairs of parentheses. If you open the brackets, moving from internal to external, then the solution will be as follows: −(−((−(5))))=−(−((−5)))=−(−(−5))=−( 5)=−5.

How to open parentheses?

Here's an explanation: −(−2 x) is +2 x, and since this expression comes first, +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 parentheses follows directly from the rule for multiplying negative numbers. Its second part is a consequence of the rule for multiplying numbers with different signs. Let's move on to examples of opening parentheses in products and quotients of two numbers with different signs.

Opening brackets: 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 parentheses in expressions that are products and partial expressions with a minus sign that are not sums and differences.

Let's look at examples of the application of this rule. Let us give the corresponding rule. Above we have already encountered expressions of the form −(a) and −(−a), which without parentheses are written as −a and a, respectively. For example, −(3)=3, and. These are special cases of the stated rule. Now let's look at examples of opening parentheses when they contain sums or differences. Let's show examples of using this rule. Let us denote the expression (b1+b2) as b, after which we use the rule of 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, in the end we get 1·3·x·y−1·2·x·y3−x·3·x·y+x·2·x·y3.

The rule in mathematics is opening parentheses if there are (+) and (-) before the brackets.

This expression is the product of three factors (2+4), 3 and (5+7·8). You will have to open the brackets 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 whose bases are some expressions written in brackets, with in kind can be thought of as the 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 a bracket by a bracket, we get a·a+a·b+a·c+b·a+b· b+b·c+c·a+c·b+c·c.

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

It remains to understand the order of opening brackets using examples. Let's take the expression (−5)+3·(−2):(−4)−6·(−7). We substitute these results into the original expression: (−5)+3·(−2):(−4)−6·(−7)=(−5)+(3·2:4)−(−6·7) . All that remains is to finish opening the brackets, as a result we have −5+3·2:4+6·7. This means that when moving from the left side of the equality to the right, the opening of the parentheses occurred.

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

How to expand parentheses to another degree

Illustrating example and rule. Let's look at an example: . You can find the value of an expression by adding 2 and 5, and then taking the resulting number from opposite sign. The rule does not change if there are not two, but three or more terms in brackets. Comment. The signs are reversed only in front of the terms. In order to open the brackets, in this case we need to remember the distributive property.

For single numbers in brackets

Your mistake is not in the signs, but in incorrect handling of fractions? In 6th grade we learned about positive and negative numbers. How will we solve examples and equations?

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

Demonstration of slide 6 with rules for opening brackets. Thus, the rules for opening parentheses will help us solve examples and simplify expressions. Next, students are asked to work in pairs: they need to use arrows to connect the expression containing brackets with the corresponding expression without brackets.

Slide 11 Once in Sunny City, Znayka and Dunno argued about which of them solved the equation correctly. Next, students solve the equation on their own using the rules for opening brackets. Solving equations” Lesson objectives: educational (reinforcement of knowledge 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 bracket, and inside these brackets the parentheses are opened according to one of the already known rules.

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Opening brackets: rules and examples (grade 7)

The main function of parentheses is to change the order of actions when calculating values numerical 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 \(5·(3+7)\) the addition in brackets will be calculated first, and only then the 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’s impossible to calculate the value in the bracket, the variable is in the way. Therefore, in this case, the brackets are “opened” using the appropriate rules.

Rules for opening parentheses

If there is a plus sign in front of 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 shorten notations, it is customary not to write the plus sign if it appears first in the expression. 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 before the bracket there is a plus, which is not written.

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

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

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

Example : Simplify the expression \(2x-(-7+x)\).
Solution : inside the bracket there are two terms: \(-7\) and \(x\), and before the bracket there is a minus. This means that the signs will change - and the seven will now be a plus, and the x will now be a minus. Open the bracket and we present similar terms .

Example. Open the bracket and give similar terms \(5-(3x+2)+(2+3x)\).
Solution : \(5-(3x+2)+(2+3x)=5-3x-2+2+3x=5\).

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 \(5(3-x)\).
Solution : In the bracket we have \(3\) and \(-x\), and before the bracket there is a five. This means that each member of the bracket is multiplied by \(5\) - I remind you that The multiplication sign between a number and a parenthesis is not written in mathematics to reduce the size of entries.

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

It remains to consider the last situation.

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

Example. Expand the brackets \((2-x)(3x-1)\).
Solution : We have a product of brackets and it can be expanded 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 and multiply each member by the second bracket:

Step 2. Expand the products of the brackets and the factor as described above:
- First things first...

Step 3. Now we multiply and present similar terms:

It is not necessary to describe all the transformations in such detail; you can multiply them right away. But if you are just learning how to open parentheses, write in detail, there will be less chance of making mistakes.

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 you substitute one instead of c, you 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 a parenthesis

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

To successfully solve such tasks, you need:
- 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 look at the task written above as an example.

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

Let's begin the task by opening the inner bracket (the one inside). Expanding it, we are dealing only with what directly relates to it - this is the bracket itself and the minus in front of it (highlighted in green). We rewrite everything else (not highlighted) the same way it was.

Solving math problems online

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Simplifying a polynomial.
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While the program is running:
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This program may be useful for students secondary schools in preparation for tests and exams, when testing knowledge before the Unified State Exam, 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 it done as quickly as possible? homework in mathematics or algebra? In this case, you can also use our programs with detailed solutions.

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A little theory.

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 terms of the polynomial. Monomials are also classified as polynomials, considering a monomial to be a polynomial consisting of one member.

Let us represent all terms in the form of monomials of the standard form:

Let us present similar terms in the resulting polynomial:

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

For degree of polynomial of a standard form take the highest of the powers of its members. Thus, a binomial has the third degree, and a trinomial has the second.

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

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

Sometimes the terms of a polynomial need to be divided into groups, enclosing each group in parentheses. Since bracketing is the inverse transformation of opening brackets, it is easy to formulate rules for opening brackets:

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

If a “-” sign is placed before the brackets, then the terms enclosed in the brackets are written with opposite signs.

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

Using the distributive property of multiplication, you 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, you must multiply that monomial by each of the terms of the polynomial.

We have already used this rule several times to multiply by a sum.

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 the following rule is used.

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 squares, differences and difference of squares

You have to deal with some expressions in algebraic transformations more often than others. Perhaps the most common expressions are u, i.e. the square of the sum, the square of the difference and the difference of squares. You noticed that the names of these expressions seem to be incomplete, for example, this is, of course, 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 does not occur very often; as a rule, instead of the letters a and b, it contains various, sometimes quite complex, expressions.

Expressions can be easily converted (simplified) into polynomials of the standard form; in fact, you have already encountered such a task when multiplying polynomials:

It is useful to remember the resulting identities and apply them without intermediate calculations. Brief verbal formulations help this.

- the square of the sum is equal to the sum of the squares and the double 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 and 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 is to see the corresponding expressions and understand how the variables a and b are replaced in them. Let's look at several examples of using abbreviated multiplication formulas.

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Expanding parentheses

We continue to study the basics of algebra. IN this lesson we will learn how to expand parentheses in expressions. Expanding parentheses means removing the parentheses from an expression.

To open parentheses, you need to memorize only two rules. With regular practice, you can open the brackets with your eyes closed, and those rules that were required to be learned by heart can be safely forgotten.

The first rule for opening parentheses

Consider the following expression:

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

The first rule for opening parentheses is as follows:

When opening brackets, if there is a plus in front of 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 sign before the parentheses. 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 without changes:

8−9+3 . This expression equals 2 , like the previous expression with brackets, was equal to 2 .

8+(−9+3) And 8−9+3

8 + (−9 + 3) = 8 − 9 + 3

Example 2. Expand parentheses in expression 3 + (−1 − 4)

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

3 + (−1 − 4) = 3 − 1 − 4

Example 3. Expand parentheses in expression 2 + (−1)

IN in this example opening the parentheses became a kind of reverse operation of replacing subtraction with addition. How to understand this?

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

Therefore, the first rule for opening parentheses can be used to simplify expressions after some transformations. That is, rid it of brackets and make it simpler.

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

To simplify this expression, similar terms can be given. Let us recall that to reduce similar terms, you need to add the coefficients of similar terms and multiply the result by the common letter part:

Got an expression 3a+(−4b). Let's remove the parentheses in this expression. There is a plus in front of 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 simplifies to 3a−4b .

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

There are two places where you need to open the parentheses. In this case, the first rule of opening parentheses applies, namely, omitting the parentheses along with the plus sign that precedes these parentheses:

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

Example 3. Expand parentheses in expression 6+(−3)+(−2)

In both places where there are parentheses, they are preceded by a plus. Here again the first rule of opening parentheses applies:

Sometimes the first term in parentheses 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 appear in front of the two 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 two.

In fact, even being in parentheses there is a plus in front of the two, but we don’t see it due to the fact that it is not written down. We have already said that the complete notation of positive numbers looks like +1, +2, +3. But according to tradition, pluses are not written down, which is why we see the positive numbers that are familiar to us 1, 2, 3 .

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

1 + (2 + 3 − 4) = 1 + 2 + 3 − 4

Example 4. Expand parentheses in 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 we write in parentheses with a plus sign:

−5 + (2 − 3) = −5 + 2 − 3

Example 5. Expand parentheses in expression (−5)

There is a plus in front of the parentheses, but it is not written down because there were no other numbers or expressions before it. Our task is to remove the parentheses by applying the first rule of opening parentheses, namely, omit the parentheses along with this plus (even if it is invisible)

Example 6. Expand parentheses in expression 2a + (−6a + b)

There is a plus in front of the brackets, which means 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 parentheses in expression 5a + (−7b + 6c) + 3a + (−2d)

There are two places in this expression where you need to expand the parentheses. In both sections there is a plus before the brackets, which means 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 rule for opening parentheses. 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 parentheses in the following expression

We see that there is a minus before the brackets. This means that you need to apply the second expansion rule, namely, omit the brackets along with the minus sign 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 parentheses 5+2+3 . This expression is equal to 10, just like the previous expression with brackets was equal to 10.

Thus, between the 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 parentheses in 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, we write the terms that were in brackets with opposite signs:

6 − (−2 − 5) = 6 + 2 + 5

Example 3. Expand parentheses in expression 2 − (7 + 3)

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

Example 4. Expand parentheses in expression −(−3 + 4)

Example 5. Expand parentheses in expression −(−8 − 2) + 16 + (−9 − 2)

There are two places where you need to open the parentheses. In the first case, you need to apply the second rule for opening parentheses, and when it 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 parentheses in expression −(−a − 1)

Example 7. Expand parentheses in expression −(4a + 3)

Example 8. Expand parentheses in expression a − (4b + 3) + 15

Example 9. Expand parentheses in expression 2a + (3b − b) − (3c + 5)

There are two places where you need to open the parentheses. In the first case, you need to apply the first rule for opening parentheses, and when it 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 parentheses in expression −a − (−4a) + (−6b) − (−8c + 15)

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

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

Bracket opening mechanism

The rules for opening brackets that we have now examined are based on the distributive law of multiplication:

In fact opening parentheses is the procedure where the common factor is multiplied by each term in parentheses. As a result of this multiplication, the brackets disappear. For example, let's expand the parentheses 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 let's open the brackets.

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

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

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

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

We expanded the brackets using the rule for expanding brackets. But these same brackets can be opened using the distributive law of multiplication. To do this, first write before the brackets the common factor 1, which was not written down:

The minus sign that previously stood before the brackets referred to this unit. Now you can open the brackets using the distributive law of multiplication. For this purpose the common factor −1 you need to multiply by each term in brackets and add the results.

For convenience, we replace the difference in parentheses with the amount:

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

Like last time we received the expression −3b+1. Everyone will agree that this time more time was spent solving such a simple example. Therefore, it is wiser to use ready-made rules opening the parentheses we looked at 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 brackets and bringing similar terms, you can slightly expand the range of problems to be solved. For example:

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

1) Open the brackets:

2) We present similar terms:

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

Example 2. Open the parentheses and add similar terms in the following expression:

1) Let's open the brackets:

2) Let us 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 an expression 8m+3m and find its value at m=−4

1) First, let's simplify the expression. To simplify the expression 8m+3m, you can take out the common factor in it m outside of brackets:

2) Find the value of the expression m(8+3) at m=−4. To do 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

“Opening parentheses” - Mathematics textbook, grade 6 (Vilenkin)

Brief description:

In this section you will learn how to expand parentheses in examples. What is this for? Everything is for the same thing as before - to make it easier and simpler for you to count, to make fewer mistakes, and ideally (the dream of your mathematics teacher) in order to solve everything without mistakes.
You already know that parentheses are placed in mathematical notation if there are two in a row mathematical sign, if we want to show the combination of numbers, their regrouping. Expanding parentheses means getting rid of unnecessary characters. For example: (-15)+3=-15+3=-12, 18+(-16)=18-16=2. Do you remember the distributive property of multiplication relative to addition? Indeed, in that example we also got rid of brackets to simplify calculations. The named property of multiplication can also be applied to four, three, five or more terms. For example: 15*(3+8+9+6)=15*3+15*8+15*9+15*6=390. Have you noticed that when you open the brackets, the numbers in them do not change sign if the number in front of the brackets is positive? After all, fifteen is a positive number. And if you solve this example: -15*(3+8+9+6)=-15*3+(-15)*8+(-15)*9+(-15)*6=-45+(- 120)+(-135)+(-90)=-45-120-135-90=-390. We had a negative number minus fifteen in front of the brackets, when we opened the brackets all the numbers began to change their sign to another - the opposite - from plus to minus.
Based on the above examples, two basic rules for opening parentheses can be stated:
1. If you have a positive number in front of the brackets, then after opening the brackets all the signs of the numbers in the brackets do not change, but remain exactly the same as they were.
2. If you have a negative number in front of the brackets, then after opening the brackets the minus sign is no longer written, and the signs of all absolute numbers in the brackets suddenly change to the opposite.
For example: (13+8)+(9-8)=13+8+9-8=22; (13+8)-(9-8)=13+8-9+8=20. Let's complicate our examples a little: (13+8)+2(9-8)=13+8+2*9-2*8=21+18-16=23. You noticed that when opening the second brackets, we multiplied by 2, but the signs remained the same as they were. Here’s an example: (3+8)-2*(9-8)=3+8-2*9+2*8=11-18+16=9, in this example the number two is negative, it’s before the brackets stands with a minus sign, so when opening them, we changed the signs of the numbers to the opposite ones (nine was with a plus, became a minus, eight was with a minus, became a plus).

That part of the equation is the expression in parentheses. To open parentheses, look at the sign in front of the parentheses. If there is a plus sign, opening the parentheses in the expression will not change anything: just remove the parentheses. If there is a minus sign, when opening the brackets, you must change all the signs that were originally in the brackets to the opposite ones. For example, -(2x-3)=-2x+3.

Multiplying two parentheses.
If the equation contains the product of two brackets, expand the brackets according to the standard rule. Each term in the first bracket is multiplied with each term in the second bracket. The resulting numbers are summed up. In this case, the product of two “pluses” or two “minuses” gives the term a “plus” sign, and if the factors have different signs, then receives a minus sign.
Let's consider.
(5x+1)(3x-4)=5x*3x-5x*4+1*3x-1*4=15x^2-20x+3x-4=15x^2-17x-4.

By opening parentheses, sometimes raising an expression to . The formulas for squaring and cubed must be known by heart and remembered.
(a+b)^2=a^2+2ab+b^2
(a-b)^2=a^2-2ab+b^2
(a+b)^3=a^3+3a^2*b+3ab^2+b^3
(a-b)^3=a^3-3a^2*b+3ab^2-b^3
Formulas for constructing expressions greater than three can be done using Pascal's triangle.

Sources:

• parenthesis expansion formula

Mathematical operations enclosed in parentheses can contain variables and expressions varying degrees complexity. To multiply such expressions, you will have to look for a solution in general view, opening the brackets and simplifying the result. If the brackets contain operations without variables, only with numerical values, then it is not necessary to open the brackets, since if you have a computer, its user has access to very significant computing resources - it is easier to use them than to simplify the expression.

Instructions

Multiply sequentially each (or minuend with ) contained in one bracket by the contents of all other brackets if you want to get the result in general form. For example, let the original expression be written as follows: (5+x)∗(6-x)∗(x+2). Then sequential multiplication (that is, opening the parentheses) will give the following result: (5+x)∗(6-x)∗(x+2) = (5∗6-5∗x)∗(5∗x+5∗2) + (6∗x-x∗x)∗(x∗x+2∗x) = (5∗6∗5∗x+5∗6∗5∗2) - (5∗x∗5∗x+5∗ x∗5∗2) + (6∗x∗x∗x+6∗x∗2∗x) - (x∗x∗x∗x+x∗x∗2∗x) = 5∗6∗5∗x + 5∗6∗5∗2 - 5∗x∗5∗x - 5∗x∗5∗2 + 6∗x∗x∗x + 6∗x∗2∗x - x∗x∗x∗x - x ∗x∗2∗x = 150∗x + 300 - 25∗x² - 50∗x + 6∗x³ + 12∗x² - x∗x³ - 2∗x³.

Simplify the result by shortening the expressions. For example, the expression obtained in the previous step can be simplified as follows: 150∗x + 300 - 25∗x² - 50∗x + 6∗x³ + 12∗x² - x∗x³ - 2∗x³ = 100∗x + 300 - 13∗ x² - 8∗x³ - x∗x³.

Use a calculator if you need to multiply x equals 4.75, that is (5+4.75)∗(6-4.75)∗(4.75+2). To calculate this value, go to the Google or Nigma search engine website and enter the expression in the query field in its original form (5+4.75)*(6-4.75)*(4.75+2). Google will show 82.265625 immediately, without pressing a button, but Nigma needs to send data to the server by pressing a button.

The main function of parentheses 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 \(5·(3+7)\) the addition in brackets will be calculated first, and only then the multiplication: \(5·(3+7)=5·10=50\).

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

Example. Open the bracket and give similar terms \(5-(3x+2)+(2+3x)\).
Solution : \(5-(3x+2)+(2+3x)=5-3x-2+2+3x=5\).

Example. Expand the brackets \(5(3-x)\).
Solution : In the bracket we have \(3\) and \(-x\), and before the bracket there is a five. This means that each member of the bracket is multiplied by \(5\) - I remind you that The multiplication sign between a number and a parenthesis is not written in mathematics to reduce the size of entries.

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

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

It remains to consider the last situation.

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

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

Example. Expand the brackets \((2-x)(3x-1)\).
Solution : We have a product of brackets and it can be expanded 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 - multiply each member by the second bracket:

Step 2. Expand the products of the brackets and the factor as described above:
- First things first...

Then the second.

Step 3. Now we multiply and present similar terms:

It is not necessary to describe all the transformations in such detail; you can multiply them right away. But if you are just learning how to open parentheses, write in detail, there will be less chance of making mistakes.

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 you substitute one instead of c, you 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 a parenthesis

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

To successfully solve such tasks, you need:
- 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 look at the task written above as an example.

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

Example. Open the brackets and give similar terms \(-(x+3(2x-1+(x-5)))\).
Solution :

\(-(x+3(2x-1\)\(+(x-5)\) \())\)		There is triple nesting of parentheses here. Let's start with the innermost one (highlighted in green). There is a plus in front of the bracket, so it simply comes off.
\(-(x+3(2x-1\)\(+x-5\) \())\)		Now you need to open the second bracket, the intermediate one. But before that, we will simplify the expression of the ghost-like terms in this second bracket.
\(=-(x\)\(+3(3x-6)\) \()=\)		Now we open the second bracket (highlighted in blue). Before the bracket is a factor - so each term in the bracket is multiplied by it.
\(=-(x\)\(+9x-18\) \()=\)
		And open the last bracket. There is a minus sign in front of the bracket, so all signs are reversed.

Expanding parentheses is a basic skill in mathematics. Without this skill, it is impossible to have a grade above a C in 8th and 9th grade. Therefore, I recommend that you understand this topic well.