Among various expressions, which are considered in algebra, important place are sums of monomials. 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.

Behind 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

With some expressions in algebraic transformations have to deal with more 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.

\((a + b)^2 = a^2 + b^2 + 2ab \) - sum squared is equal to the sum squares and 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.

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

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

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

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

How to open brackets?

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

Bracket opening: rules, examples, solutions.

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

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

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

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

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

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

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

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

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

How to open parentheses in a different degree

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

Single numbers in brackets

Your error lies not in the signs, but in wrong work with fractions? In the 6th grade, we met with positive and negative numbers. How will we solve examples and equations?

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

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

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

Lesson topic: “Opening parentheses. In this case, you need to multiply each term from the first brackets with each term from the second brackets and then add the results. First, the first two factors are taken, enclosed in one more brackets, and inside these brackets, the brackets are opened according to one of the already known rules.

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Bracket opening: rules and examples (Grade 7)

The main function of brackets is to change the order of actions when calculating values numeric expressions . For example, V in numerical terms\(5 3+7\) multiplication will be calculated first, and then addition: \(5 3+7 =15+7=22\). But in the expression \(5·(3+7)\), addition in brackets will be calculated first, and only then multiplication: \(5·(3+7)=5·10=50\).

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

Bracket expansion rules

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

Here it is necessary to clarify that in mathematics, to reduce entries, it is customary not to write the plus sign if it is the first in the expression. For example, if we add two positive numbers, for example, seven and three, then we do not write \(+7+3\), but simply \(7+3\), despite the fact that seven is also positive number. Similarly, if you see, for example, the expression \((5+x)\) - know that there is a plus in front of the bracket, which is not written.

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

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

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

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

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

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

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

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

It remains to consider the last situation.

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

Example. Expand the brackets \((2-x)(3x-1)\).
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. We remove the first bracket - each of its members is multiplied by the second bracket:

Step 2. Expand the products of the bracket by the factor as described above:
- the first one first...

Step 3. Now we multiply and bring like terms:

It is not necessary to paint all the transformations in detail, you can immediately multiply. But if you are just learning to open brackets - write in detail, there will be less chance of making a mistake.

Note to the entire section. In fact, you don't need to remember all four rules, you only need to remember one, this one: \(c(a-b)=ca-cb\) . Why? Because if we substitute one instead of c, we get the rule \((a-b)=a-b\) . And if we substitute minus one, we get the rule \(-(a-b)=-a+b\) . Well, if you substitute another bracket instead of c, you can get the last rule.

parenthesis within parenthesis

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

To be successful in these tasks, you need to:
- carefully understand the nesting of brackets - which one is in which;
- open the brackets sequentially, starting, for example, with the innermost one.

It is important when opening one of the brackets don't touch the rest of the expression, just rewriting it as is.
Let's take the task above as an example.

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

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

Solving problems in mathematics online

Online calculator.
Polynomial simplification.
Multiplication of polynomials.

With this math program, you can simplify a polynomial.
While the program is running:
- multiplies polynomials
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- opens brackets
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The polynomial simplification program does not just give the answer to the problem, it leads detailed solution with explanations, i.e. displays the solution process so that you can check your knowledge of mathematics and / or algebra.

This program may be useful for students general education schools in preparation for control work 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 get it done as soon as possible? homework math or algebra? In this case, you can also use our programs with a detailed solution.

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Because There are a lot of people who want to solve the problem, your request is queued.
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A bit of theory.

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

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

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

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

We give similar terms in the resulting polynomial:

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

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

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

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

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

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

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

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

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

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

This result is usually formulated as a rule.

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

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

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

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

Usually use the following rule.

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

Abbreviated multiplication formulas. Sum, Difference, and Difference Squares

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

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

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

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

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

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

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

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Bracket expansion

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

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

The first rule of parenthesis expansion

Consider the following expression:

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

The first parenthesis expansion rule looks like this:

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

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

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

8+(−9+3) And 8−9+3

8 + (−9 + 3) = 8 − 9 + 3

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

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

3 + (−1 − 4) = 3 − 1 − 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1 + (2 + 3 − 4) = 1 + 2 + 3 − 4

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

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

−5 + (2 − 3) = −5 + 2 − 3

Example 5 Expand brackets in an expression (−5)

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

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

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

2a + (−6a + b) = 2a −6a + b

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

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

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

The second rule for opening parentheses

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

If there is a minus before the brackets, then this minus is omitted along with the brackets, but the terms that were in the brackets change their sign to the opposite.

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

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

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

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

5 − (−2 − 3) = 5 + 2 + 3

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

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

6 − (−2 − 5) = 6 + 2 + 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Parentheses expansion mechanism

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

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

3 × (4 + 5) = 3 × 4 + 3 × 5

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

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

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

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

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

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

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

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

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

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

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

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

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

1) Expand the brackets:

2) We give like terms:

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

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

1) Expand the brackets:

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

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

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

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

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

In this article, we will take a closer look at the basic rules of such important topic course of mathematics, as the opening of brackets. You need to know the rules for opening brackets in order to correctly solve equations in which they are used.

How to properly open parentheses when adding

Expand the brackets preceded by the "+" sign

This is the simplest case, because if there is an addition sign in front of the brackets, when the brackets are opened, the signs inside them do not change. Example:

(9 + 3) + (1 - 6 + 9) = 9 + 3 + 1 - 6 + 9 = 16.

How to open brackets preceded by a "-" sign

In this case, you need to rewrite all the terms without brackets, but at the same time change all the signs inside them to the opposite ones. The signs change only for the terms from those brackets that were preceded by the “-” sign. Example:

(9 + 3) - (1 - 6 + 9) = 9 + 3 - 1 + 6 - 9 = 8.

How to open 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 signs. If the multiplier has the sign "-", then when multiplying, the signs of the terms are reversed. Example:

3 * (1 - 6 + 9) = 3 * 1 - 3 * 6 + 3 * 9 = 3 - 18 + 27 = 12.

How to open two brackets with a multiplication sign between them

In this case, you need to multiply each term from the first brackets with each term from the second brackets and then add the results. Example:

(9 + 3) * (1 - 6 + 9) = 9 * 1 + 9 * (- 6) + 9 * 9 + 3 * 1 + 3 * (- 6) + 3 * 9 = 9 - 54 + 81 + 3 - 18 + 27 = 48.

How to open brackets in a square

If the sum or difference of two terms is squared, the brackets should be expanded according to the following formula:

(x + y)^2 = x^2 + 2*x*y + y^2.

In the case of a minus inside the brackets, the formula does not change. Example:

(9 + 3) ^ 2 = 9 ^ 2 + 2 * 9 * 3 + 3 ^ 2 = 144.

How to open parentheses in a different degree

If the sum or difference of the terms is raised, for example, to the 3rd or 4th power, then you just need to break the degree of the bracket into “squares”. The powers of the same factors are added, and when dividing, the degree of the divisor is subtracted from the degree of the dividend. Example:

(9 + 3) ^ 3 = ((9 + 3) ^ 2) * (9 + 3) = (9 ^ 2 + 2 * 9 * 3 + 3 ^ 2) * 12 = 1728.

How to open 3 brackets

There are equations in which 3 brackets are multiplied at once. In this case, you must first multiply the terms of the first two brackets among themselves, and then multiply the sum of this multiplication by the terms of the third bracket. Example:

(1 + 2) * (3 + 4) * (5 - 6) = (3 + 4 + 6 + 8) * (5 - 6) = - 21.

These bracket opening rules apply equally to both linear and trigonometric equations.

Parentheses are used to indicate the order in which operations are performed in numerical and literal 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 works are expanded in accordance with distributive property 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

I continue a series of methodological articles on the topic of teaching. It's time to consider the features individual work math tutor with 7th grade students. With great pleasure I will share my thoughts on the forms of submission of one of the major topics algebra course in grade 7 - "opening brackets." In order not to try to embrace the immensity, let's focus on her elementary school and analyze the methodology of the tutor with the multiplication of a polynomial by a polynomial. How math tutor valid in difficult situations, When weak student does not perceive classic shape explanations? What tasks should be prepared for a strong seventh grader? Let's consider these and other questions.

It would seem, well, what's so difficult? “Parentheses are easy,” any good student will say. “There is a distributive law and properties of degrees for working with monomials, a general algorithm for any number of terms. Multiply each by each and bring the like. However, not everything is so simple in working with the lagging behind. Despite the efforts of a math tutor, students manage to make mistakes of various calibers even in the simplest transformations. The nature of the errors is striking in its diversity: from small omissions of letters and signs, to serious dead-end "stop errors".

What prevents the student from correctly performing the transformations? Why is there misunderstanding?

There are individual problems great multitude and one of the main obstacles to the assimilation and consolidation of the material is the difficulty in timely and quick switching of attention, the difficulty in processing a large amount of information. It may seem strange to some that I am talking about large volume, but a weak student of grade 7 may not have enough memory and attention resources even for four terms. Coefficients, variables, degrees (indicators) interfere. The student confuses the sequence of operations, forgets which monomials have already been multiplied and which have remained untouched, cannot remember how they are multiplied, etc.

Math tutor's numerical approach

Of course, you need to start with an explanation of the logic of building the algorithm itself. How to do it? We need to set the task: how to change the order of actions in the expression without changing the result? I quite often give examples explaining the operation of certain rules on specific numbers. And then I replace them with letters. The technique for using the numerical approach will be described below.

Problems of motivation.
At the beginning of the lesson, it is difficult for a math tutor to gather a student if he does not understand the relevance of what is being studied. Within the framework of the program for grades 6-7, it is difficult to find examples of using the polynomial multiplication rule. I would emphasize the need to learn change the order of actions in expressions The fact that this helps to solve problems, the student should know from the experience of adding similar terms. He also had to add them in when solving equations. For example, in 2x+5x+13=34 he uses that 2x+5x=7x. A math tutor just needs to focus the attention of the student on this.

Mathematics teachers often call the parenthesis opening technique fountain rule.

This image is well remembered and must be used. But how is this rule proven? Recall the classical form using obvious identity transformations:

(a+b)(c+d)=(a+b) c+(a+b) d=ac+bc+ad+bd

It is difficult for a math tutor to comment on anything here. The letters speak for themselves. Yes, and not needed by a strong student of grade 7 detailed explanations. However, what to do with the weak, who point-blank does not see any content in this "alphabetic mishmash"?

The main problem that hinders the perception of the classical mathematical justification of the "fountain" is the unusual form of writing the first factor. Neither in the 5th grade nor in the 6th grade did the student have to drag the first bracket to each term of the second. Children dealt only with numbers (coefficients), located, most often, to the left of the brackets, for example:

By the end of the 6th grade, the student develops visual image object - a certain combination of signs (actions) associated with brackets. And any deviation from the usual look towards something new can disorient a seventh grader. It is the visual image of the “number + bracket” pair that the math tutor takes into circulation when explaining.

The following explanation can be offered. The tutor argues: “If there was some number in front of the bracket, for example 5, then we could change the course of action in this expression? Certainly. Let's do it then . Think about whether its result will change if instead of the number 5 we enter the sum of 2 + 3 enclosed in brackets? Any student will tell the tutor: "What difference does it make how to write: 5 or 2 + 3." Wonderful. Get a record. The math tutor takes a short pause so that the student visually remembers the picture-image of the object. Then he draws his attention to the fact that the bracket, like the number, "distributed" or "jumped" to each term. What does this mean? It means that this operation can be performed not only with a number, but also with a bracket. We got two pairs of factors and . With them most of students can easily cope on their own and write out the result to the tutor. It is important to compare the resulting pairs with the content of brackets 2+3 and 6+4 and it will become clear how they open.

If necessary, after the example with numbers, the math tutor conducts a literal proof. It turns out to be a cakewalk through the same parts of the previous algorithm.

Formation of the skill of opening brackets

The formation of the skill of multiplying brackets is one of milestones work of a tutor in mathematics with a theme. And even more important than the stage of explaining the logic of the “fountain” rule. Why? The justifications for the transformations will be forgotten the very next day, and the skill, if it is formed and fixed in time, will remain. Students perform the operation mechanically, as if extracting the multiplication table from memory. This is what needs to be achieved. Why? If every time the student opens the brackets, he will remember why he opens it this way and not otherwise, he will forget about the problem he is solving. That is why the math tutor spends the rest of the lesson on transforming understanding into rote memorization. This strategy is often used in other topics as well.

How can a tutor develop the skill of opening brackets in a student? To do this, a 7th grade student must perform a series of exercises in sufficient quantity to consolidate. This raises another problem. A weak seventh grader cannot cope with the increased number of transformations. Even small ones. And mistakes keep coming in one after the other. What should a math tutor do? First, it is necessary to recommend painting arrows from each term to each. If the student is very weak and is not able to quickly switch from one type of work to another, loses concentration when following simple commands from the teacher, then the math tutor draws these arrows himself. And not all at once. First, the tutor connects the first term of the left bracket with each term of the right bracket and asks to perform the appropriate multiplication. Only after that do the arrows go from the second term to the same right bracket. In other words, the tutor divides the process into two stages. It is better to maintain a small temporary pause (5-7 seconds) between the first and second operation.

1) One set of arrows should be drawn above the expressions and another set below them.
2) It is important to skip between lines at least couple of cells. Otherwise, the record will be very dense, and the arrows will not only climb to the previous line, but will also mix with the arrows from the next exercise.

3) In the case of multiplying brackets in the format 3 by 2, arrows are drawn from the short bracket to the long one. Otherwise, these "fountains" will be not two, but three. The implementation of the third is noticeably more complicated due to the lack of free space for the arrows.
4) the arrows are always directed from one point. One of my students kept trying to put them side by side and this is what he did:

Such an arrangement does not allow to single out and fix the current term, with which the student works at each of the stages.

The work of the tutor's fingers

4) To keep attention on a separate couple multiplied terms, the math tutor puts two fingers on them. This must be done in such a way as not to block the student's view. For the most inattentive students, you can use the "pulsation" method. The math tutor brings the first finger to the beginning of the arrow (to one of the terms) and fixes it, and with the second “knocks” on its end (on the second term). Pulsation helps to focus attention on the term by which the student multiplies. After the first multiplication by the right bracket is done, the math tutor says: “Now we work with another term.” The tutor moves a “fixed finger” to it, and “pulsating” runs over the terms from another bracket. The pulsation works like a "turn signal" in a car and allows you to collect the attention of an absent-minded student on the operation he is conducting. If the child writes small, then two pencils are used instead of fingers.

Repetition optimization

As in the study of any other topic in the course of algebra, the multiplication of polynomials can and should be integrated with previously covered material. To do this, the math tutor uses special bridge tasks that allow you to find the application of the studied in various mathematical objects. They not only connect the topics into a single whole, but also very effectively organize the repetition of the entire course of mathematics. And the more bridges the tutor builds, the better.

Traditionally, in algebra textbooks for grade 7, bracket opening is integrated with the solution of linear equations. At the end of the list of numbers there are always tasks of the following order: solve the equation. When opening the brackets, the squares are reduced and the equation is easily solved by means of class 7. However, for some reason, the authors of textbooks safely forget about plotting a graph of a linear function. In order to correct this shortcoming, I would advise math tutors to include brackets in analytic expressions linear functions, For example . On such exercises, the student not only trains the skills of carrying out identical transformations, but also repeats the graphs. You can ask to find the intersection point of two "monsters", determine mutual arrangement lines, find the points of their intersection with the axes, etc.

Kolpakov A.N. Mathematics tutor in Strogino. Moscow