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

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

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

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

Bracket expansion rule for addition

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

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

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

The rule for expanding brackets when subtracting

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

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

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

2 − (7 + 3) = 2 − (+ 7 + 3)

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

2 − (+ 7 + 3) = 2 − 7 − 3

Expanding parentheses when multiplying

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

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

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

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

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

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

Expand parentheses when dividing

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

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

How to expand nested parentheses

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

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

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

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 form.

• 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 text, see How to Use Quotations in Text Properly.

In this lesson, you will learn how to transform an expression that contains parentheses into an expression that does not contain parentheses. You will learn how to open brackets 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 linking new and previously studied material into a single whole.

Topic: Equation Solving

Lesson: Parentheses expansion

How to open brackets preceded by a "+" sign. Use of the associative law of addition.

If you need to add the sum of two numbers to a number, then you can 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 passing from the left side of the equality to the right side, the brackets were opened.

Consider examples.

Example 1

Expanding the brackets, we changed the order of operations. Counting has become more convenient.

Example 2

Example 3

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

Comment.

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

You can follow the step by step example. 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.

If you follow the indicated order of actions, then you must first subtract 345 from 512, and then add 1345 to the result. By expanding the brackets, we will change the order of actions and greatly simplify the calculations.

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. We get -7.

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

Let's formulate the 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. 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.

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 is preceded by a “-” sign, therefore, all signs must be reversed

Bibliography

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. Tasks for the course of mathematics grade 5-6 - ZSH MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - ZSH MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of high school. Library of the teacher of mathematics. - Enlightenment, 1989.

1. Online math tests ().
2. You can download the ones 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. (see link 1.2)
2. Homework: No. 1254, No. 1255, No. 1256 (b, d)
3. Other assignments: No. 1258(c), No. 1248

Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:
\(5a^4 - 2a^3 + 0.3a^2 - 4.6a + 8 \)
\(xy^3 - 5x^2y + 9x^3 - 7y^2 + 6x + 5y - 2 \)

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.

For example, polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.

We represent all the terms as monomials of the standard form:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16 \)

We give similar terms in the resulting polynomial:
\(8b^5 -14b^5 +3b^2 -8b -3b^2 + 16 = -6b^5 -8b + 16 \)
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.

Per polynomial degree standard form take the largest of the powers of its members. So, the binomial \(12a^2b - 7b \) has the third degree, and the trinomial \(2b^2 -7b + 6 \) has the second.

Usually, the terms of standard form polynomials containing one variable are arranged in descending order of its exponents. For example:
\(5x - 18x^3 + 1 + x^5 = x^5 - 18x^3 + 5x + 1 \)

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:
\(9a^2b(7a^2 - 5ab - 4b^2) = \)
\(= 9a^2b \cdot 7a^2 + 9a^2b \cdot (-5ab) + 9a^2b \cdot (-4b^2) = \)
\(= 63a^4b - 45a^3b^2 - 36a^2b^3 \)

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 \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), that is, the square of the sum, the square of the difference, and square difference. You have noticed that the names of these expressions seem to be incomplete, so, for example, \((a + b)^2 \) 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 is not so common, as a rule, instead of the letters a and b, it contains various, sometimes quite complex expressions.

Expressions \((a + b)^2, \; (a - b)^2 \) are easy to convert (simplify) into polynomials of the standard form, in fact, you have already met with such a task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)

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

\((a + b)^2 = a^2 + b^2 + 2ab \) - the square of the sum is equal to the sum of the squares and the double product.

\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is the sum of the squares without doubling the product.

\(a^2 - b^2 = (a - b)(a + b) \) - 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 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.

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: \(-(4m+3)\).
Solution : \(-(4m+3)=-4m-3\).

Example. Expand the bracket and give like 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 : 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 \(-2(-3x+5)\).
Solution : As in the previous example, the bracketed \(-3x\) and \(5\) 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 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 \((2-x)(3x-1)\).
Solution : 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 \(7x+2(5-(3x+y))\).
Solution:

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

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