In this article, we will consider in detail the basic rules of such an important topic of the course of mathematics as the disclosure of brackets. You need to know the rules for opening brackets in order to correctly solve the equations in which they are used.
How to open brackets when adding
We open the brackets before which the “+” sign stands
This is the simplest case, because if the addition sign is in front of the brackets, the signs inside them do not change when the brackets are opened. Example:
(9 + 3) + (1 - 6 + 9) = 9 + 3 + 1 - 6 + 9 = 16.
How to open brackets preceded by a “-” sign
In this case, you need to rewrite all the terms without brackets, but at the same time change all the signs inside them to the opposite. Signs change only in terms of those brackets in front of which there was a “-" sign. Example:
(9 + 3) - (1 - 6 + 9) = 9 + 3 - 1 + 6 - 9 = 8.
How to expand parentheses when multiplying
The parentheses are preceded by a multiplier
In this case, you need to multiply each term by a factor and open the brackets without changing the signs. If the factor has a “-” sign, then when multiplying the signs of the terms change to opposite. Example:
3 * (1 - 6 + 9) = 3 * 1 - 3 * 6 + 3 * 9 = 3 - 18 + 27 = 12.
How to open two brackets with a multiplication sign between them
In this case, it is necessary to multiply each term from the first brackets with each term from the second brackets and then add the results. Example:
(9 + 3) * (1 - 6 + 9) = 9 * 1 + 9 * (- 6) + 9 * 9 + 3 * 1 + 3 * (- 6) + 3 * 9 = 9 - 54 + 81 + 3 - 18 + 27 = 48.
How to expand brackets in a square
If the sum or difference of the two terms is squared, the brackets should be disclosed according to the following formula:
(x + y) ^ 2 \u003d x ^ 2 + 2 * x * y + y ^ 2.
In the case of a minus, the formula does not change inside the brackets. Example:
(9 + 3) ^ 2 = 9 ^ 2 + 2 * 9 * 3 + 3 ^ 2 = 144.
How to expand brackets to another degree
If the sum or difference of terms is raised, for example, to the 3rd or 4th degree, then you just need to divide the degree of the bracket into “squares”. The degrees of the same factors are added, and when dividing, the degree of the divisor is subtracted from the degree of the dividend. Example:
(9 + 3) ^ 3 = ((9 + 3) ^ 2) * (9 + 3) = (9 ^ 2 + 2 * 9 * 3 + 3 ^ 2) * 12 = 1728.
How to expand 3 brackets
There are equations in which 3 brackets are multiplied at once. In this case, you must first multiply the terms of the first two brackets, and then multiply the sum of this multiplication by the terms of the third bracket. Example:
(1 + 2) * (3 + 4) * (5 - 6) = (3 + 4 + 6 + 8) * (5 - 6) = - 21.
These rules for opening brackets are equally applicable for solving both linear and trigonometric equations.
Everywhere. Everywhere and wherever you look, these structures are found here:
These "constructions" in literate people cause a mixed reaction. At least the type of "is it really right?"
In general, I personally can’t understand where the “fashion” came from, not to close the external quotation marks. The first and only analogy that comes about this is that of parentheses. No one doubts that two brackets in a row is normal. For example: "Pay for the entire circulation (200 pcs. (Of which 100 - marriage))." But someone doubted the normality of putting two quotes in a row (I wonder who is the first?) ... And now, all without a doubt, they began to produce designs like LLC Pupkov & Co. Firm.
But even if you haven’t seen the rule in your life, which will be discussed below, the only logical option (using parentheses as an example) would be the following: Firm Pupkov & Co LLC.
So, the rule itself:
If at the beginning or at the end of a quotation (the same applies to direct speech) there are internal and external quotes, then they must differ in the pattern (the so-called "Christmas trees" and "paws"), and external quotes should not be omitted, for example: the sides of the ship were broadcast on the radio: “Leningrad entered the tropics and continues on its course.” Belinsky writes about Zhukovsky: “Contemporaries of Zhukovsky’s youth looked at him mainly as the author of ballads, and in one of his letters Batyushkov called him“ ballad ”.”
© Rules of Russian spelling and punctuation. - Tula: Autograph, 1995 .-- 192 p.
Accordingly ... if you do not have the opportunity to type quotes, "Christmas tree", then what can you do, you will have to use such "" icons. However, the inability (or reluctance) to use Russian quotation marks is by no means a reason why external quotation marks cannot be closed.Thus, the incorrect design of LLC “Firm Pupkov & Co” seems to have been sorted out. There are also constructions of the form LLC “Firm Pupkov & Co”.
It’s completely clear from the rule that such constructions are illiterate ... (Correct: LLC “Pupkov & Co.” Ltd.)However!
In the “Handbook of the publisher and author” A. E. Milchin (2004 edition) it is indicated that two design options can be used in such cases. The use of "herringbone" and "legs" and (in the absence of technical means) the use of only "herringbone": two opening and one closing.
The reference book is “fresh” and personally I immediately have 2 questions. Firstly, with what joy it is possible to use one closing Christmas tree quote (well, this is illogical, see above), and secondly, the phrase “in the absence of technical means” is especially noteworthy. How's that, sorry? So open Notepad and type “only Christmas trees: two opening and one closing” there. There are no such characters on the keyboard. It doesn’t work to print the Christmas tree ... The combination of Shift + 2 gives the sign "(which, as you know, is not a quotation mark). Now open Microsoft Word and press Shift + 2 again. The program will fix" on (or) ) Well, it turns out that the rule that existed for more than a dozen years was taken and rewritten under Microsoft Word? Like, since a Word from "Pupkov & Co. Firm" is being done by "Pupkov & Co. Firm, then now let it be permissible 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 Windows computer there are always “technical means” for entering both “Christmas trees” and “paws”, so this new “rule” (for me it is in quotation marks) is incorrect from the very beginning!
All special font characters can be easily typed knowing the corresponding number of this character. Just hold down Alt and type on the NumLock keyboard (NumLock is pressed, the indicator light is on) the corresponding character number:
„Alt + 0132 (left“ foot ”)
“Alt + 0147 (right foot”)
"Alt + 0171 (left" Christmas tree ")
»Alt + 0187 (right" herringbone ")
The main function of the brackets is to change the order of operations in the calculation of values. for example, in the numerical expression \\ (5 · 3 + 7 \\), the multiplication will be calculated first, and then the addition: \\ (5 · 3 + 7 \u003d 15 + 7 \u003d 22 \\). But in the expression \\ (5 · (3 + 7) \\), the addition in parenthesis will be calculated first, and only then the multiplication: \\ (5 · (3 + 7) \u003d 5 · 10 \u003d 50 \\).
Example.
Expand the bracket: \\ (- (4m + 3) \\).
Decision
: \\ (- (4m + 3) \u003d - 4m-3 \\).
Example.
Expand the bracket and give similar terms \\ (5- (3x + 2) + (2 + 3x) \\).
Decision
: \\ (5- (3x + 2) + (2 + 3x) \u003d 5-3x-2 + 2 + 3x \u003d 5 \\).
Example.
Expand the brackets \\ (5 (3-x) \\).
Decision
: In the bracket we have \\ (3 \\) and \\ (- x \\), and in front of the bracket there are five. So, each member of the bracket is multiplied by \\ (5 \\) - I remind you that the multiplication sign between a number and a bracket is not written in mathematics to reduce the size of records.
Example.
Expand the brackets \\ (- 2 (-3x + 5) \\).
Decision
: As in the previous example, the brackets \\ (- 3x \\) and \\ (5 \\) are multiplied by \\ (- 2 \\).
Example.
Simplify the expression: \\ (5 (x + y) -2 (x-y) \\).
Decision
: \\ (5 (x + y) -2 (x-y) \u003d 5x + 5y-2x + 2y \u003d 3x + 7y \\).
It remains to consider the last situation.
When multiplying a bracket by a bracket, each member of the first bracket is multiplied with each member of the second:
\\ ((c + d) (a-b) \u003d c (a-b) + d (a-b) \u003d ca-cb + da-db \\)
Example.
Expand the brackets \\ ((2-x) (3x-1) \\).
Decision
: We have a product of brackets and it can be opened immediately by the formula above. But in order not to get confused, let's do everything in steps.
Step 1. We remove the first bracket - each of its member is multiplied by the second bracket:
Step 2. We open the product of the bracket by the factor as described above:
- first first ...
Then the second.
Step 3. Now we multiply and present similar terms:
It’s not necessary to paint all the transformations in detail, you can immediately multiply. But if you are only learning to open brackets - write in detail, there will be less chance of mistakes.
Note to the entire section. In fact, you don’t need to remember all four rules, it’s enough to remember only one, this: \\ (c (a-b) \u003d ca-cb \\). Why? Because if you substitute one for c instead of it, you get the rule \\ ((a-b) \u003d a-b \\). And if you substitute minus one, we get the rule \\ (- (a-b) \u003d - a + b \\). Well, if you substitute another bracket for c, you can get the last rule.
Bracket in 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 successfully solve such tasks, you need:
- carefully understand the nesting of brackets - which one to be in;
- open the brackets sequentially, starting, for example, with the innermost one.
It’s important when opening one of the brackets do not touch the rest of the expressionjust rewriting it as it is.
Let's take an example of the above task.
Example.
Expand the brackets and give similar terms \\ (7x + 2 (5- (3x + y)) \\).
Decision:
Example.
Expand the brackets and give similar terms \\ (- (x + 3 (2x-1 + (x-5))) \\).
Decision
:
|
\\ (- (x + 3 (2x-1 \\) \\ (+ (x-5) \\) \\ ()) \\) |
Here is the triple nesting of brackets. We start from the innermost (highlighted in green). There is a plus in front of the bracket, so it is simply removable. |
|
|
\\ (- (x + 3 (2x-1 \\) \\ (+ x-5 \\) \\ ()) \\) |
Now you need to open the second bracket, intermediate. But before that we simplify a ghost expression similar to the terms in this second bracket. |
|
|
\\ (\u003d - (x \\) \\ (+ 3 (3x-6) \\) \\ () \u003d \\) |
Now we open the second bracket (highlighted in blue). There is a multiplier in front of the bracket — so each term in the bracket is multiplied by it. |
|
|
\\ (\u003d - (x \\) \\ (+ 9x-18 \\) \\ () \u003d \\) |
||
|
And open the last bracket. There is a minus in front of the bracket - therefore, all signs are reversed. |
||
Opening brackets is a basic skill in mathematics. Without this skill, it is impossible to have a rating higher than three in grades 8 and 9. Therefore, I recommend that you understand this topic well.
Now we’ll just move on to expanding the brackets in the expressions in which the expression in parentheses is multiplied by a number or expression. We formulate the rule for opening brackets before which there is a minus sign: the brackets along with the minus sign are omitted, and the signs of all terms in brackets are replaced by opposite ones.
One type of expression conversion is parenthesized expansion. Numeric, alphabetic and variable expressions are composed using brackets, which can indicate the order in which actions are performed, contain a negative number, etc. Suppose that in the expressions described above there can be any expressions instead of numbers and variables.
And let’s pay attention to one more point regarding the features of recording solutions when opening brackets. In the previous paragraph, we figured out what is called opening brackets. To do this, there are rules for opening brackets, which we begin to review. This rule is dictated by the fact that positive numbers are usually written without brackets, brackets in this case are superfluous. The expression (−3.7) - (- 2) +4 + (- 9) can be written without brackets as −3.7 + 2 + 4−9.
Finally, the third part of the rule is simply due to the peculiarities of writing negative numbers to the left of the expression (as we mentioned in the section for brackets for writing negative numbers). You may encounter expressions made up of numbers, minus signs, and several pairs of brackets. If you open the brackets, moving from internal to external, the solution will be as follows: - (- ((- (5)))) \u003d - (- ((- 5))) \u003d - (- (- 5)) \u003d - ( 5) \u003d - 5.
How to open brackets?
Here is the explanation: - (- 2 · x) is + 2 · x, and since this expression is first, then + 2 · x can be written as 2 · x, - (x2) \u003d - x2, + (- 1 / x) \u003d - 1 / x and - (2 · x · y2: z) \u003d - 2 · x · y2: z. The first part of the written parenthesis expansion rule follows directly from the rule of multiplying negative numbers. The second part is a consequence of the rule of multiplying numbers with different signs. We turn to examples of the disclosure of brackets in works and quotients of two numbers with different signs.
Disclosure of brackets: rules, examples, solutions.
The above rule takes into account the whole chain of these actions and significantly speeds up the process of opening brackets. The same rule allows you to open brackets in expressions, which are products and particular expressions with a minus sign, which are not sums and differences.
Consider examples of applying this rule. We give the corresponding rule. Above, we have already encountered expressions of the form - (a) and - (- a), which without brackets are written as −a and a, respectively. For example, - (3) \u003d 3, and. These are special cases of the stated rule. Now let's look at examples of parentheses when sums or differences are enclosed in them. We show examples of the use of this rule. Denote the expression (b1 + b2) as b, after which we use the rule of multiplying the brackets by the expression from the previous paragraph, we have (a1 + a2) · (b1 + b2) \u003d (a1 + a2) · b \u003d (a1 · b + a2 · b) \u003d a1b + a2b.
By induction, this statement can be extended to an arbitrary number of terms in each bracket. It remains to open the brackets in the expression obtained using the rules from the previous paragraphs, as a result we get 1 · 3 · x · y − 1 · 2 · x · y3 − x · 3 · x · y + x · 2 · x · y3.
The rule in mathematics is the disclosure of brackets if the brackets are preceded by (+) and (-)
This expression is the product of three factors (2 + 4), 3 and (5 + 7 · 8). Brackets will have to be opened sequentially. Now we use the rule of multiplying the brackets by a number, we have ((2 + 4) · 3) · (5 + 7 · 8) \u003d (2 · 3 + 4 · 3) · (5 + 7 · 8). Degrees based on some expressions written in brackets with natural indicators can be considered as the product of several brackets.
For example, we transform the expression (a + b + c) 2. First, we write it in the form of the product of two brackets (a + b + c) · (a + b + c), now we multiply the bracket by the bracket, we get a · a + a · b + a · c + b · a + b · b + b · c + c · a + c · b + c · c.
We also say that for raising the sums and differences of two numbers to a natural degree, it is advisable to use the Newton binomial formula. For example, (5 + 7−3): 2 \u003d 5: 2 + 7: 2−3: 2. It is equally convenient to pre-divide by replacement by multiplication, and then use the appropriate rule for opening brackets in a work.
It remains to deal with the order of disclosing the brackets in the examples. Take the expression (−5) + 3 · (−2): (- 4) −6 · (−7). We substitute these results into the original expression: (−5) + 3 · (−2): (- 4) −6 · (−7) \u003d (- 5) + (3 · 2: 4) - (- 6 · 7) . It remains only to complete the disclosure of the brackets, as a result, we have −5 + 3 · 2: 4 + 6 · 7. So, when moving from the left side of the equality to the right, the brackets were opened.
Note that in all three examples, we simply removed the brackets. First, add 445 to 889. This action can be performed in the mind, but it is not very simple. We’ll open the brackets and see that the changed procedure will greatly simplify the calculation.
How to expand brackets to another degree
Illustrative example and rule. Consider an example:. You can find the value of the expression by adding 2 and 5, and then take the resulting number with the opposite sign. A rule does not change if there are not two, but three or more terms in brackets. Comment. Signs are reversed only before the terms. In order to open the brackets, in this case we need to recall the distribution property.
Single numbers in brackets
Your mistake is not in signs, but in the incorrect work with fractions? In grade 6, we met with positive and negative numbers. How will we solve examples and equations?
How much did it happen in parentheses? What can be said about these expressions? Of course, the result of the first and second examples is the same, so you can put an equal sign between them: -7 + (3 + 4) \u003d -7 + 3 + 4. What did we do with the brackets?
Demonstration of a slide 6 with rules of disclosure of brackets. Thus, the rules for opening brackets will help us solve examples and simplify expressions. Next, students are encouraged to work in pairs: you need to use an arrow to connect an expression containing brackets with the corresponding expression without brackets.
Slide 11 Once in the Solar City, Znayka and Dunno argued which of them solved the equation correctly. Next, students independently solve the equation, using the rules for opening brackets. Solving Equations ”Lesson objectives: educational (fixing ZUNs on the topic:“ Opening brackets.
Theme of the lesson: “Disclosure of brackets. In this case, it is necessary 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 parenthesis, and inside these brackets the brackets are opened according to one of the already known rules.
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Disclosure of brackets: rules and examples (Grade 7)
The main function of the brackets is to change the order in the calculation of values numeric expressions . for example, in the numerical expression \\ (5 · 3 + 7 \\), the multiplication will be calculated first, and then the addition: \\ (5 · 3 + 7 \u003d 15 + 7 \u003d 22 \\). But in the expression \\ (5 · (3 + 7) \\), the addition in parenthesis will be calculated first, and only then the multiplication: \\ (5 · (3 + 7) \u003d 5 · 10 \u003d 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 interferes. Therefore, in this case, the brackets are “opened” using the appropriate rules for this.
Parenthesis disclosure rules
If the plus sign is 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 reduce records, it is customary not to write a plus sign if it is the first in the expression. For example, if we add two positive numbers, for example, seven and three, then we 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 in front of the bracket is a plus that they don’t write.
Example
. Expand the bracket and give similar terms: \\ ((x-11) + (2 + 3x) \\).
Decision
: \\ ((x-11) + (2 + 3x) \u003d x-11 + 2 + 3x \u003d 4x-9 \\).
If there is a minus sign in front of the bracket, then when removing the bracket, each member of the expression inside it changes the sign to the opposite:
Here it is necessary to clarify that a, while it was in the bracket, had a plus sign (they simply did not write it), and after removing the bracket, this plus changed to minus.
Example
: Simplify the expression \\ (2x - (- 7 + x) \\).
Decision
: there are two terms inside the bracket: \\ (- 7 \\) and \\ (x \\), and minus in front of the bracket. So, the signs will change - and the seven will now be with a plus, and X - with a minus. We open the bracket and we give similar terms .
Example.
Expand the bracket and give similar terms \\ (5- (3x + 2) + (2 + 3x) \\).
Decision
: \\ (5- (3x + 2) + (2 + 3x) \u003d 5-3x-2 + 2 + 3x \u003d 5 \\).
If there is a factor in front of the bracket, then each member of the bracket is multiplied by it, that is:
Example.
Expand the brackets \\ (5 (3-x) \\).
Decision
: In the bracket we have \\ (3 \\) and \\ (- x \\), and in front of the bracket there are five. So, each member of the bracket is multiplied by \\ (5 \\) - I remind you that the multiplication sign between a number and a bracket is not written in mathematics to reduce the size of records.
Example.
Expand the brackets \\ (- 2 (-3x + 5) \\).
Decision
: As in the previous example, the brackets \\ (- 3x \\) and \\ (5 \\) are multiplied by \\ (- 2 \\).
It remains to consider the last situation.
When multiplying a bracket by a bracket, each member of the first bracket is multiplied with each member of the second:
Example.
Expand the brackets \\ ((2-x) (3x-1) \\).
Decision
: We have a product of brackets and it can be opened immediately by the formula above. But in order not to get confused, let's do everything in steps.
Step 1. We remove the first bracket - each of its member is multiplied by the second bracket:
Step 2. We open the product of the bracket by the factor as described above:
- first first ...
Step 3. Now we multiply and present similar terms:
It’s not necessary to paint all the transformations in detail, you can immediately multiply. But if you are only learning to open brackets - write in detail, there will be less chance of mistakes.
Note to the entire section. In fact, you don’t need to remember all four rules, it’s enough to remember only one, this: \\ (c (a-b) \u003d ca-cb \\). Why? Because if you substitute one for c instead of it, you get the rule \\ ((a-b) \u003d a-b \\). And if you substitute minus one, we get the rule \\ (- (a-b) \u003d - a + b \\). Well, if you substitute another bracket for c, you can get the last rule.
Bracket in 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 successfully solve such tasks, you need:
- carefully understand the nesting of brackets - which one to be in;
- open the brackets sequentially, starting, for example, with the innermost one.
It’s important when opening one of the brackets do not touch the rest of the expressionjust rewriting it as it is.
Let's take an example of the above task.
Example.
Expand the brackets and give similar terms \\ (7x + 2 (5- (3x + y)) \\).
Decision:
We start the task by opening the inner bracket (the one inside). Opening it, we deal only with the fact that it is directly related to it - this is the bracket itself and the minus sign in front of it (highlighted in green). We rewrite the rest (not selected) as it was.
Solving math problems online
Online calculator.
Simplification of the polynomial.
Multiplication of polynomials.
With this math program, you can simplify a polynomial.
In the process, the program:
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The polynomial simplification program not only gives an answer to the problem, it provides a detailed solution with explanations, i.e. displays the decision process so that you can test your knowledge of mathematics and / or algebra.
This program can be useful to students of secondary schools in preparation for tests and exams, when testing knowledge before the exam, 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 do your homework in math or algebra as quickly as possible? In this case, you can also use our programs with a detailed solution.
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A bit of theory.
The product of the monomial and polynomial. Polynomial concept
Among the various expressions that are considered in algebra, an important place is occupied by the sum of monomials. We give examples of such expressions:
The sum of monomials is called a polynomial. The terms in the polynomial are called members of the polynomial. Monomials are also referred to as polynomials, considering the monomial to be a polynomial consisting of one member.
We represent all terms in the form of monomials of the standard form:
We give similar terms in the resulting polynomial:
The result is a polynomial, all members of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.
Behind polynomial degree standard form take the largest of the degrees of its members. So, the binomial has the third degree, and the trinomial - the second.
Usually members of polynomials of the standard form containing one variable are arranged in decreasing order of exponents of its degree. For example:
The sum of several polynomials can be converted (simplified) into a polynomial of standard form.
Sometimes members of a polynomial need to be divided into groups, enclosing each group in brackets. Since bracketing is the opposite of brackets, it’s easy to formulate parenthesis disclosure 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 the monomial and polynomial
Using the distributive property of multiplication, one can transform (simplify) a product of a monomial and a polynomial into a polynomial. For example:
The product of the monomial and polynomial is identically equal to the sum of the products of this monomial and each of the members of the polynomial.
This result is usually formulated as a rule.
To multiply a monomial by a polynomial, you must multiply this monomial by each member of the polynomial.
We have repeatedly used this rule to multiply by the 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 member of one polynomial and each member of the other.
Usually use the following rule.
To multiply a polynomial by a polynomial, it is necessary to multiply each member of one polynomial by each member of the other and add the resulting products.
Abbreviated Multiplication Formulas. Squares of the sum, difference and difference of squares
Some expressions in algebraic transformations have to be dealt with more often than others. Perhaps the most common expression and, i.e., the square of the sum, the square of the difference and the difference of squares. You noticed that the names of the indicated expressions are not finished, so, for example, this, of course, is not just a square of the sum, but a 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 rather complex expressions.
It is easy to transform (simplify) the expressions into polynomials of the standard form; in fact, you have already encountered this task when multiplying polynomials:
It is useful to remember and apply the obtained identities without intermediate calculations. Short 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 a double product.
- the difference of the squares is equal to the product of the difference by the sum.
These three identities make it possible in transformations to replace their left parts with the right and vice versa - the right parts with the left. 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 a few examples of using abbreviated multiplication formulas.
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Parenthesis
We continue to study the basics of algebra. In this lesson, we will learn to open brackets in expressions. Expanding the brackets means removing the expression from these brackets.
To open the brackets, you need to memorize just two rules. With regular classes, you can open the brackets with your eyes closed, and those rules that needed to be memorized can be safely forgotten.
The first rule for expanding parentheses
Consider the following expression:
The value of this expression is 2 . Expand the brackets in this expression. Opening brackets 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 expanding brackets is as follows:
When opening brackets, if a plus is in front of the brackets, then this plus is omitted along with the brackets.
So we see that in the expression 8+(−9+3) in front of the brackets is a plus. This plus should be omitted with brackets. In other words, the brackets will disappear along with the plus that stood in front of them. And what was in brackets will be written unchanged:
8−9+3 . This expression is equal to 2 as the previous expression with brackets was equal 2 .
8+(−9+3) and 8−9+3
8 + (−9 + 3) = 8 − 9 + 3
Example 2 Expand brackets in expression 3 + (−1 − 4)
In front of the brackets is a plus, so this plus is omitted along with the brackets. What was in brackets will remain unchanged:
3 + (−1 − 4) = 3 − 1 − 4
Example 3 Expand brackets in expression 2 + (−1)
In this example, the disclosure of the brackets became a kind of inverse operation, replacing the subtraction by addition. What does it mean?
In expression 2−1 a 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 expanding brackets can be used to simplify expressions after some transformations. That is, to remove it from brackets and make it easier.
For example, we simplify the expression 2a + a − 5b + b .
To simplify this expression, we can cite similar terms. Recall that to reduce such terms, you need to add the coefficients of such terms and multiply the result by the total letter part:
Got expression 3a + (- 4b) . In this expression, we expand the brackets. The brackets are preceded by a plus, so we use the first rule for expanding brackets, that is, we omit the brackets along with the plus that comes before these brackets:
So the expression 2a + a − 5b + b simplified to 3a − 4b .
Having opened some brackets, others may meet along the way. We apply the same rules to them as to the first. For example, we expand the brackets in the following expression:
There are two places where you need to expand the brackets. In this case, the first rule for opening brackets is applicable, namely, omitting the brackets along with the plus that stands before these brackets:
2 + (−3 + 1) + 3 + (−6) = 2 − 3 + 1 + 3 − 6
Example 3 Expand brackets in expression 6+(−3)+(−2)
In both places where there are brackets, there is a plus in front of them. Here again, the first rule for expanding brackets applies:
Sometimes the first term in parentheses is written without sign. For example, in the expression 1+(2+3−4) first term in brackets 2 recorded without a sign. The question arises, which character will stand in front of the deuce after the brackets and the plus in front of the brackets go down? The answer suggests itself - a plus will stand in front of a deuce.
In fact, even being in brackets in front of the deuce is a plus, but we do not see it because it is not recorded. We have already said that a complete record of positive numbers looks like +1, +2, +3. But traditionally they don’t write the pluses, that's why we see the positive numbers that are familiar to us 1, 2, 3 .
Therefore, to expand the brackets in the expression 1+(2+3−4) , you need to omit the brackets as usual, along with the plus in front of these brackets, but write the first term in brackets with a plus sign:
1 + (2 + 3 − 4) = 1 + 2 + 3 − 4
Example 4 Expand brackets in expression −5 + (2 − 3)
The brackets are preceded by a plus, so we apply the first rule for opening brackets, namely, we omit the brackets along with the plus that stands before these brackets. But the first term, which is written in brackets with a plus sign:
−5 + (2 − 3) = −5 + 2 − 3
Example 5 Expand brackets in expression (−5)
The parenthesis is preceded by a plus, but it is not written down because there were no other numbers or expressions before it. Our task is to remove the brackets by applying the first rule of disclosing the brackets, namely, omit the brackets along with this plus (even if it is invisible)
Example 6 Expand brackets in expression 2a + (−6a + b)
In front of the brackets is a plus, so this plus is omitted along with the brackets. What was in brackets will be written unchanged:
2a + (−6a + b) \u003d 2a −6a + b
Example 7 Expand brackets in expression 5a + (−7b + 6c) + 3a + (−2d)
There are two places in this expression where you need to expand the brackets. In both sections, a plus is in front of the brackets, so this plus is omitted along with the brackets. What was in brackets will be written unchanged:
5a + (−7b + 6c) + 3a + (−2d) \u003d 5a −7b + 6c + 3a - 2d
The second parenthesis rule
Now consider the second rule for expanding brackets. It is used when the minus sign is in front of the brackets.
If there is a minus in front of 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, we expand the brackets in the following expression
We see that in front of the brackets is a minus. So you need to apply the second disclosure rule, namely, omit the brackets along with the minus in front of these brackets. In this case, the terms that were in brackets will change their sign to the opposite:
We got an expression without brackets 5+2+3 . This expression is 10, like the previous expression with brackets was 10.
So between expressions 5−(−2−3) and 5+2+3 You can put an equal sign, because they are equal to the same value:
5 − (−2 − 3) = 5 + 2 + 3
Example 2 Expand brackets in expression 6 − (−2 − 5)
The minus sign is in front of the brackets, therefore, we apply the second rule for opening the brackets, namely, we omit the brackets along with the minus sign in front of these brackets. In this case, the terms that were in parentheses are written with opposite signs:
6 − (−2 − 5) = 6 + 2 + 5
Example 3 Expand brackets in expression 2 − (7 + 3)
The minus sign is in front of the brackets, so we use the second rule for opening brackets:
Example 4 Expand brackets in expression −(−3 + 4)
Example 5 Expand brackets in expression −(−8 − 2) + 16 + (−9 − 2)
There are two places where you need to expand the brackets. In the first case, you need to apply the second rule for opening brackets, and when the queue reaches the expression +(−9−2) you need to apply the first rule:
−(−8 − 2) + 16 + (−9 − 2) = 8 + 2 + 16 − 9 − 2
Example 6 Expand brackets in expression - (- a - 1)
Example 7 Expand brackets in expression - (4a + 3)
Example 8 Expand brackets in expression a - (4b + 3) + 15
Example 9 Expand brackets in expression 2a + (3b - b) - (3c + 5)
There are two places where you need to expand the brackets. In the first case, you need to apply the first rule for opening brackets, and when the queue reaches the expression - (3c + 5) you need to apply the second rule:
2a + (3b - b) - (3c + 5) = 2a + 3b - b - 3c - 5
Example 10 Expand brackets in expression −a - (−4a) + (−6b) - (−8c + 15)
There are three places where you need to open the brackets. First, you need to apply the second parenthesis disclosure rule, then the first, and then the second again:
−a - (−4a) + (−6b) - (−8c + 15) = −a + 4a - 6b + 8c - 15
Parenthesis disclosure mechanism
The brackets disclosure rules that we have just reviewed are based on the distribution law of multiplication:
In fact parenthesis call the procedure when the common factor is multiplied by each term in brackets. As a result of such a multiplication, the brackets disappear. For example, expand the brackets in the expression 3 × (4 + 5)
3 × (4 + 5) \u003d 3 × 4 + 3 × 5
Therefore, if you need to multiply the number by the expression in parentheses (or the expression in parentheses by the number), you need to say open brackets.
But how is the distribution law of multiplication connected with the rules for opening brackets that we considered earlier?
The fact is that before any brackets there is a common factor. In the example 3 × (4 + 5) common factor is 3 . And in the example a (b + c) common factor is a variable a.
If there are no numbers or variables in front of the brackets, then the common factor is 1 or −1 , depending on which character is in front of the brackets. If there is a plus in front of the brackets, then the common factor is 1 . If there is a minus in front of the brackets, then the common factor is −1 .
For example, expand the brackets in the expression - (3b − 1) . The brackets are preceded by a minus, so you need to use the second rule for opening brackets, that is, omit the brackets along with the minus in front of the brackets. And the expression that was in brackets is written with opposite signs:
We opened the brackets using the parenthesis disclosure rule. But these same brackets can be opened using the distribution law of multiplication. To do this, first write in front of the brackets the common factor 1, which was not written:
The minus that used to stand in front of the brackets applied to this unit. Now you can open the brackets using the distribution law of multiplication. For this, the common factor −1 you need to multiply by each term in brackets and add the results.
For convenience, replace the difference in parentheses by the amount:
−1 (3b −1) \u003d −1 (3b + (−1)) \u003d −1 × 3b + (−1) × (−1) \u003d −3b + 1
Like last time we got the expression −3b + 1 . Everyone will agree that this time spent more time on solving such a simple example. Therefore, it is wiser to use ready-made rules for opening brackets, which we examined in this lesson:
But it does not bother to know how these rules work.
In this lesson, we have learned yet another identical transformation. Together with the disclosure of the brackets, putting the common out of the brackets and bringing such terms, it is possible to slightly expand the range of tasks. For example:
Here you need to perform two actions - first, open the brackets, and then give similar terms. So, in order:
1) We open the brackets:
2) We give similar terms:
In the resulting expression −10b + (- 1) brackets can be expanded:
Example 2 Expand the brackets and give similar terms in the following expression:
1) Expand the brackets:
2) We give similar terms. This time, to save time and space, we will not write how the coefficients are multiplied by the total letter part
Example 3 Simplify expression 8m + 3m and find its value when m \u003d −4
1) First, simplify the expression. To simplify the expression 8m + 3m , you can take out the common factor in it m out of brackets:
2) Find the value of the expression m (8 + 3) at m \u003d −4 . For this in expression m (8 + 3) instead of variable m substitute the number −4
m (8 + 3) \u003d −4 (8 + 3) \u003d −4 × 8 + (−4) × 3 \u003d −32 + (−12) \u003d −44
A + (b + c) can be written without brackets: a + (b + c) \u003d a + b + c. This operation is called bracketing.
Example 1We expand the brackets in the expression a + (- b + c).
Decision. a + (-b + c) \u003d a + ((-b) + c) \u003d a + (-b) + c \u003d a-b + c.
If the “+” sign is in front of the brackets, then you can omit the brackets and this “+” sign while retaining the signs of the terms in brackets. If the first term in brackets is written without a sign, then it must be written with a “+” sign.
Example 2 Find the value of the expression -2.87+ (2.87-7.639).
Decision. Opening the brackets, we get - 2.87 + (2.87 - 7.639) \u003d - - 2.87 + 2.87 - 7.639 \u003d 0 - 7.639 \u003d - 7.639.
To find the value of the expression - (- 9 + 5), we must add the numbers -9 and 5 and find the number opposite to the sum obtained: - (- 9 + 5) \u003d - (- 4) \u003d 4.
The same value can be obtained in a different way: first write down the numbers opposite to the given terms (that is, change their signs), and then add: 9 + (- 5) \u003d 4. Thus, - (- 9 + 5) \u003d 9 - 5 \u003d 4.
To write down the sum opposite to the sum of several terms, it is necessary to change the signs of these terms.
Therefore, - (a + b) \u003d - a - b.
Example 3 Find the value of the expression 16 - (10 -18 + 12).
Decision. 16-(10 -18 + 12) = 16 + (-(10 -18 + 12)) = = 16 + (-10 +18-12) = 16-10 +18-12 = 12.
To open the brackets before the “-” sign, you need to replace this sign with a “+”, changing the signs of all terms in brackets to the opposite, and then open the brackets.
Example 4 Find the value of the expression 9.36- (9.36 - 5.48).
Decision. 9.36 - (9.36 - 5.48) \u003d 9.36 + (- 9.36 + 5.48) \u003d 9.36 - 9.36 + 5.48 \u003d 0 -f 5.48 \u003d 5 , 48.
Brackets and translational and combinational properties additions simplify calculations.
Example 5 Find the value of the expression (-4-20) + (6 + 13) - (7-8) -5.
Decision. First, we open the brackets, and then we find separately 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 6Find the value of the expression
Decision.First, we represent each term in the form of the sum of their integer and fractional parts, then we expand the brackets, then we add separately the whole and separately fractional parts and finally add up the results:
How do parentheses preceded by a “+” sign open? How can I find the value of an expression opposite to the sum of several numbers? How to open brackets preceded by a “-" sign?
1218. Expand 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. Expand the brackets:
a) 85+ (7.8+ 98); d) - (80-16) + 84; g) a- (b-k-n);
b) (4.7 -17) +7.5; d) -a + (m-2.6); h) - (a-b + c);
c) 64- (90 + 100); e) c + (- a-b); i) (m-n) - (p-k).
1221. Expand the brackets and find the value of the expression:
1222. Simplify the expression:
1223. Write the 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; d) - m + n and -k - n;
c) a + 13 and -13 + b; f) m - n and n - m.
1224. Write the difference of the two expressions and simplify it:
1226. Using the equation, 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 as were left on the second. After that, 12 books remained on the first shelf. How many books were removed from the second shelf?
b) In the first grade there are 42 students, in the second 3 students less than in the third. How many students are in the third grade, if there are 125 students in all three of these classes?
1227. Find the meaning of the expression:
1228. Calculate orally:
1229. Find the largest value of the expression:
1230. Indicate 4 consecutive integers if:
a) the smaller of them is -12; c) the smaller of them is equal to n;
b) the larger of them is -18; d) the larger of them is equal to k.