Online calculator. Simplifying a polynomial. Multiplying polynomials. How to use simple parentheses

Expanding parentheses is a type of expression transformation. In this section we will describe the rules for opening parentheses, and also look at the most common examples of problems.

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What is opening parentheses?

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. For example, replace the expression 2 · (3 + 4) with an expression of the form 2 3 + 2 4 without parentheses. This technique is called opening brackets.

Definition 1

Expanding parentheses refers to techniques for getting rid of parentheses and is usually considered in relation to expressions that may contain:

• signs “+” or “-” before parentheses containing 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 are used to considering the process of opening brackets in the course school curriculum. However, no one is stopping us from looking at this action more broadly. We can call parenthesis opening 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 an opening of parentheses.

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 opening parentheses.

Here's another example. We can assume that any expressions can be used in expressions instead of numbers and variables. 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 recording decisions 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.

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 parentheses, examples

Let's start looking at the rules for opening parentheses.

For single numbers in brackets

Negative numbers in parentheses are often found in expressions. For example, (− 4) and 3 + (− 4) . Positive numbers in brackets also have a place.

Let us formulate a rule for opening parentheses containing single positive numbers. Let's assume that 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 , 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 , because + (− 5) is replaced by − 5 .

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

Now consider the rule for opening parentheses 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 (− 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 rules for opening parentheses 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 create 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 subtraction rules negative numbers we can state that − (− a) = a, a − (− b) = a + b.

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

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

For example, after opening the parentheses the expression − (− 2 x) − (x 2) + (− 1 x) − (2 x y 2: z) will take 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 products of two numbers

Let's start with the rule for opening parentheses in the product of two numbers.

Let's assume that a and b is two positive numbers. In this case, the product of two negative numbers − a and − b of the form (− a) · (− b) we can replace with (a · b) , and the products of two numbers with opposite signs of the form (− a) · b and a · (− b) can be replaced with (− 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

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

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

In place of negative numbers − a and − b can be any expressions with a minus sign in front that are not sums or differences. For example, these can be products, quotients, fractions, powers, 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 into the expression (− 3 2) . After this you can expand the brackets: − 3 2.

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

Dividing numbers with different signs may also require preliminary expansion of parentheses: (− 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) = (- sin (x) x 2) = - sin (x) x 2

In products of three or more numbers

Let's move on to the products and quotients that contain more numbers. To open brackets, the following rule will apply here. At even number For negative numbers, you can omit the parentheses and replace the numbers with their opposites. After this, you need to enclose the resulting expression in new brackets. If there is an odd number of negative numbers, omit the parentheses and replace the numbers with their opposites. After this, the resulting expression must be placed in new brackets and a minus sign must be placed in front of it.

Example 2

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

In the product (− 2, 5) · (− 3) : (− 2) · 4: (− 1, 25) : (− 1) five numbers are negative. therefore (− 2, 5) · (− 3) : (− 2) · 4: (− 1, 25) : (− 1) = (− 2, 5 · 3: 2 · 4: 1, 25: 1) . Having finally opened the brackets, we get −2.5 3:2 4:1.25:1.

The above rule can be justified as follows. Firstly, we can rewrite such expressions as a product, replacing them with multiplication by reciprocal number division. We represent each negative number as the product of a multiplying number and - 1 or - 1 is replaced by (− 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 one is equal to 1, and the product of 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 to open the parentheses 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 opening parentheses in expressions that represent products and quotients with a minus sign that are not sums or differences. Let's take for example the expression

x 2 · (- x) : (- 1 x) · x - 3: 2 .

It can be reduced to the expression without parentheses x 2 · x: 1 x · x - 3: 2.

Expanding parentheses preceded by a + sign

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

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

Example 3

For example, we give the expression (12 − 3 , 5) − 7 . By omitting the parentheses, we keep the signs of the terms in parentheses 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 example given, it is not necessary to place a sign in front of the first term, since + 12 − 3, 5 − 7 = 12 − 3, 5 − 7.

Example 4

Let's look at another example. Let's take the expression x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x and carry out the 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's 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 + x 2

How are parentheses preceded by a minus sign expanded?

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

Example 6

For example:

1 2 = 1 2 , - 1 x + 1 = - 1 x + 1 , - (- x 2) = x 2

Expressions with variables 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 look at cases where you need to expand parentheses that are multiplied or divided by some number or expression. 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 parentheses in the expression (3 − 7) 2. According to the rule, we can carry out the following transformations: (3 − 7) · 2 = (3 · 2 − 7 · 2) . We get 3 · 2 − 7 · 2 .

Opening the parentheses 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.

Multiplying parenthesis by parenthesis

Consider the product of two brackets of the form (a 1 + a 2) · (b 1 + b 2) . This will help us obtain a rule for opening parentheses when performing bracket-by-bracket multiplication.

In order to solve the given example, we denote the expression (b 1 + b 2) like b. This will allow us to use the rule for multiplying a parenthesis by an expression. 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 performing a reverse replacement b by (b 1 + b 2), again apply the rule of multiplying an expression by a 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 techniques, we can arrive at the sum of the products of each of the terms from the first bracket by 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: to multiply two sums together, you need 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 the 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

It is worth mentioning separately those cases where there is a minus sign in parentheses along with plus signs. For example, take the expression (1 − x) · (3 · x · y − 2 · x · y 3) .

First, let's present 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 open the brackets: 1 · 3 · x · y − 1 · 2 · x · y 3 − x · 3 · x · y + x · 2 · x · y 3 .

Expanding parentheses in products of multiple parentheses and expressions

If there are three or more expressions in parentheses in an expression, the parentheses must be opened sequentially. You need to start the transformation by putting the first two factors in brackets. Within these brackets we can carry out 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 open the brackets sequentially. Let's enclose the first two factors in one more bracket, which we'll make red for clarity: (2 + 4) 3 (5 + 7 8) = ((2 + 4) 3) (5 + 7 8).

In accordance with the rule for 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 .

Bracket in kind

Degrees, the bases of which are some expressions written in brackets, with natural exponents can be considered as the 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 the product of two brackets (a + b + c) · (a + b + c). Let's 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 look at 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 parenthesis by number and parentheses by parenthesis

Dividing a bracket by a number requires that all terms enclosed in brackets be divided by the number. For example, (x 2 - x) : 4 = x 2: 4 - x: 4 .

Division can first be replaced by multiplication, after which you can use the appropriate rule for opening parentheses in a product. The same rule applies when dividing a parenthesis by a parenthesis.

For example, we need to open the parentheses in the expression (x + 2) : 2 3 . To do this, first replace division by multiplying by the reciprocal number (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's another example of division by parenthesis:

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 .

Order of opening brackets

Now let’s consider the order of application of the rules discussed above in general expressions, i.e. in expressions that contain sums with differences, products with quotients, parentheses to the natural degree.

Procedure:

• the first step is to raise the brackets to a natural power;
• at the second stage, the opening of brackets in works and quotients is carried out;
• The final step is to open the parentheses 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) . When substituting the obtained results into the original expression, we obtain: (− 5) + 3 · (− 2) : (− 4) − 6 · (− 7) = (− 5) + (3 · 2: 4) − (− 6 · 7). Open the brackets: − 5 + 3 · 2: 4 + 6 · 7.

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

If you notice an error in the text, please highlight it and press Ctrl+Enter

A+(b + c) can be written without parentheses: a+(b + c)=a + b + c. This operation is called opening parentheses.

Example 1. Let's open the brackets in the expression a + (- b + c).

Solution. a + (-b+c) = a + ((-b) + c)=a + (-b) + c = a-b + c.

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

Example 2. Let's find the value of the expression -2.87+ (2.87-7.639).

Solution. Opening the brackets, we get - 2.87 + (2.87 - 7.639) = - - 2.87 + 2.87 - 7.639 = 0 - 7.639 = - 7.639.

To find the value of the expression - (- 9 + 5), you need to add numbers-9 and 5 and find the number opposite to the resulting sum: -(- 9 + 5)= -(- 4) = 4.

The same value can be obtained in another way: first write down the numbers opposite to these terms (i.e. change their signs), and then add: 9 + (- 5) = 4. Thus, -(- 9 + 5) = 9 - 5 = 4.

To write a sum opposite to the sum of several terms, you need to change the signs of these terms.

This means - (a + b) = - a - b.

Example 3. Let's find the value of the expression 16 - (10 -18 + 12).

Solution. 16-(10 -18 + 12) = 16 + (-(10 -18 + 12)) = = 16 + (-10 +18-12) = 16-10 +18-12 = 12.

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

Example 4. Let's find the value of the expression 9.36-(9.36 - 5.48).

Solution. 9.36 - (9.36 - 5.48) = 9.36 + (- 9.36 + 5.48) = = 9.36 - 9.36 + 5.48 = 0 -f 5.48 = 5 ,48.

Expanding parentheses and applying commutative and associative properties addition allow you to simplify calculations.

Example 5. Let's find the value of the expression (-4-20)+(6+13)-(7-8)-5.

Solution. First, we will open the brackets, and then we will separately find the sum of all positive and separately the sum of all negative numbers and, finally, add up the results:

(- 4 - 20)+(6+ 13)-(7 - 8) - 5 = -4-20 + 6 + 13-7 + 8-5 = = (6 + 13 + 8)+(- 4 - 20 - 7 - 5)= 27-36=-9.

Example 6. Let's find the value of the expression

Solution. First, let’s imagine each term as the sum of their integer and fractional parts, then open the brackets, then add the integers and separately fractional parts and finally add up the results:

How do you open parentheses preceded by a “+” sign? How can you find the value of an expression that is the opposite of the sum of several numbers? How to expand parentheses preceded by a “-” sign?

1218. Open the brackets:

a) 3.4+(2.6+ 8.3); c) m+(n-k);

b) 4.57+(2.6 - 4.57); d) c+(-a + b).

1219. Find the meaning of the expression:

1220. Open the brackets:

a) 85+(7.8+ 98); d) -(80-16) + 84; g) a-(b-k-n);
b) (4.7 -17)+7.5; e) -a + (m-2.6); h) -(a-b + c);
c) 64-(90 + 100); e) c+(- a-b); i) (m-n)-(p-k).

1221. Open the brackets and find the meaning of the expression:

1222. Simplify the expression:

1223. Write amount two expressions and simplify it:

a) - 4 - m and m + 6.4; d) a+b and p - b
b) 1.1+a and -26-a; e) - m + n and -k - n;
c) a + 13 and -13 + b; e)m - n and n - m.

1224. Write the difference of two expressions and simplify it:

1226. Use the equation to solve the problem:

a) There are 42 books on one shelf, and 34 on the other. Several books were removed from the second shelf, and as many books were taken from the first shelf as were left on the second. After that, there were 12 books left on the first shelf. How many books were removed from the second shelf?

b) There are 42 students in the first grade, 3 students less in the second than in the third. How many students are there in third grade if there are 125 students in these three grades?

1227. Find the meaning of the expression:

1228. Calculate orally:

1229. Find highest value expressions:

1230. Specify 4 consecutive integers if:

a) the smaller of them is -12; c) the smaller of them is n;
b) the largest of them is -18; d) the greater of them is equal to k.

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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 like this: (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 clicking a button, but Nigma needs to send data to the server with a click of a button.

In this lesson you will learn how to transform an expression containing parentheses into an expression without parentheses. You will learn how to open parentheses preceded by a plus sign and a minus sign. We will remember how to open brackets using the distributive law of multiplication. The considered examples will allow you to connect new and previously studied material into a single whole.

Topic: Solving equations

Lesson: Expanding Parentheses

How to expand parentheses preceded by a “+” sign. Using the associative law of addition.

If you need to add the sum of two numbers to a number, you can first add the first term to this number, and then the second.

To the left of the equal sign is an expression with parentheses, and to the right is an expression without parentheses. This means that when moving from the left side of the equality to the right, the opening of the parentheses occurred.

Let's look at examples.

Example 1.

By opening the brackets, we changed the order of actions. It has become more convenient to count.

Example 2.

Example 3.

Note that in all three examples we simply removed the parentheses. Let's formulate a rule:

Comment.

If the first term in brackets is unsigned, then it must be written with a plus sign.

You can follow the example step by step. 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.

If you follow the indicated procedure, you must first subtract 345 from 512, and then add 1345 to the result. By opening the brackets, we will change the procedure and significantly simplify the calculations.

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. We get -7.

On the other hand, the same result can be obtained by adding the opposite numbers of the original ones.

Let's formulate a rule:

Example 1.

Example 2.

The rule does not change if there are not two, but three or more terms in brackets.

Example 3.

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.

First, multiply the first bracket by 2, and the second by 3.

The first bracket is preceded by a “+” sign, which means that the signs must be left unchanged. The second sign is preceded by a “-” sign, therefore, all signs need to be changed to the opposite

References

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Assignments for the mathematics course grades 5-6 - ZSh MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - ZSh MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for grades 5-6 high school. Math teacher's library. - Enlightenment, 1989.

1. Online tests in mathematics ().
2. You can download those specified in clause 1.2. books().

Homework

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012. (link see 1.2)
2. Homework: No. 1254, No. 1255, No. 1256 (b, d)
3. Other tasks: No. 1258(c), No. 1248

Everywhere. Everywhere and wherever you look, you can see these constructions:

These “constructions” cause mixed reactions among literate people. At least like “is this really correct?”
In general, I personally cannot understand where the “fashion” of not closing outer quotes came from. The first and only analogy that comes to this is the analogy with parentheses. No one doubts that two brackets in a row is normal. For example: “Pay for the entire circulation (200 pieces (of which 100 are defective)).” But someone doubted the normality of putting two quotation marks in a row (I wonder who was first?) ... And now everyone has become completely clear conscience to produce structures such as Firm Pupkov and Co. LLC.
But even if you have never seen the rule in your life, which will be discussed below, then the only logical option (using the example of brackets) would be the following: LLC Firm Pupkov and Co.
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 should differ from each other in design (the so-called “herringbones” and “petals”), and the external quotation marks should not be omitted, for example: C The sides of the steamer radioed: “Leningrad has entered the tropics and is continuing on its course.” About Zhukovsky, Belinsky writes: “Contemporaries of Zhukovsky’s youth looked at him primarily as an author of ballads, and in one of his letters Batyushkov called him a “balladeer.”
© Rules of Russian spelling and punctuation. - Tula: Autograph, 1995. - 192 p.
Accordingly... if you don’t have the opportunity to type “herringbone” quotes, then what can you do, you’ll have to use such “” icons. However, the inability (or unwillingness) to use Russian quotation marks is by no means a reason why you can not close external quotation marks.

Thus, the inaccuracy of the design of LLC "Firm Pupkov and Co" seems to have been sorted out. There are also designs of the type LLC Firm "Pupkov and Co".
It is completely clear from the rule that such constructions are also illiterate... (Correct: LLC "Firm Pupkov and Co"

However!
The Publisher's and Author's Guide by A.E. Milchin (2004 edition) states that two design options can be used in such cases. The use of “herringbones” and “legs” and (in the absence of technical means) the use of only “herringbones”: two opening ones and one closing one.
The directory is “fresh” and personally, I immediately have 2 questions here. Firstly, with what joy can one use one closing quotation mark (well, this is illogical, see above), and secondly, the phrase “in the absence of technical means” especially attracts attention. How is this, excuse me? Now open Notepad and type “only Christmas trees: two opening and one closing.” There are no such symbols on the keyboard. I can’t print “herringbone”... The combination Shift + 2 produces the sign " (which, as you know, is not 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 decades was taken and rewritten under Microsoft Word? Like, since the Word from the "Firm "Pupkov and Co" makes "Firm "Pupkov and Co", then let this now be acceptable and correct???
It seems so. And if this is 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 computer with Windows there are always “technical means” for entering both “Christmas trees” and “legs”, so this new “rule” (for me it is in quotes) is incorrect from the very beginning!

All special characters in a font can be easily typed by knowing the corresponding number of that character. Just 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)