Bracket multiplication. Bracket opening: rules and examples (Grade 7)
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 positive number. Similarly, if you see, for example, the expression (5 + x) - know that there is a plus in front of the bracket, which is not written, and there is a plus + (+5 + x) in front of the five.
Bracket expansion rule for addition
When opening brackets, if there is a plus before the brackets, then this plus is omitted along with the brackets.
Example. Open the brackets in the expression 2 + (7 + 3) Before the brackets plus, then the characters in front of the numbers in the brackets do not change.
2 + (7 + 3) = 2 + 7 + 3
The rule for expanding brackets when subtracting
If there is a minus before the brackets, then this minus is omitted along with the brackets, but the terms that were in the brackets change their sign to the opposite. The absence of a sign before the first term in parentheses implies a + sign.
Example. Open brackets in expression 2 − (7 + 3)
There is a minus before the brackets, so you need to change the signs before the numbers from the brackets. There is no sign in brackets before the number 7, which means that the seven is positive, it is considered that the + sign is in front of it.
2 − (7 + 3) = 2 − (+ 7 + 3)
When opening the brackets, we remove the minus from the example, which was before the brackets, and the brackets themselves 2 − (+ 7 + 3), and change the signs that were in the brackets to the opposite ones.
2 − (+ 7 + 3) = 2 − 7 − 3
Expanding parentheses when multiplying
If there is a multiplication sign in front of the brackets, then each number inside the brackets is multiplied by the factor in front of the brackets. At the same time, multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus, gives a minus.
Thus, parentheses in products are expanded in accordance with the distributive property of multiplication.
Example. 2 (9 - 7) = 2 9 - 2 7
When multiplying parenthesis by parenthesis, each term of the first parenthesis is multiplied with every term of the second parenthesis.
(2 + 3) (4 + 5) = 2 4 + 2 5 + 3 4 + 3 5
In fact, there is no need to remember all the rules, it is enough to remember only one, this one: c(a−b)=ca−cb. Why? Because if we substitute one instead of c, we get the rule (a−b)=a−b. And if we substitute minus one, we get the rule −(a−b)=−a+b. Well, if you substitute another bracket instead of c, you can get the last rule.
Expand parentheses when dividing
If there is a division sign after the brackets, then each number inside the brackets is divisible by the divisor after the brackets, and vice versa.
Example. (9 + 6) : 3=9: 3 + 6: 3
How to expand nested parentheses
If the expression contains nested brackets, then they are expanded in order, starting with external or internal.
At the same time, when opening one of the brackets, it is important not to touch the other brackets, just rewriting them as they are.
Example. 12 - (a + (6 - b) - 3) = 12 - a - (6 - b) + 3 = 12 - a - 6 + b + 3 = 9 - a + b
Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:
\(5a^4 - 2a^3 + 0.3a^2 - 4.6a + 8 \)
\(xy^3 - 5x^2y + 9x^3 - 7y^2 + 6x + 5y - 2 \)
The sum of monomials is called a polynomial. The terms in a polynomial are called members of the polynomial. Mononomials are also referred to as polynomials, considering a monomial as a polynomial consisting of one member.
For example, polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.
We represent all the terms in the form of monomials standard view:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16 \)
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
Some expressions in algebraic transformations have to be dealt with more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), that is, the square of the sum, the square of the difference, and square difference. You have noticed that the names of these expressions seem to be incomplete, so, for example, \((a + b)^2 \) is, of course, not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b is not so common, as a rule, instead of the letters a and b, it contains various, sometimes quite complex expressions.
Expressions \((a + b)^2, \; (a - b)^2 \) are easy to convert (simplify) into polynomials of the standard form, in fact, you have already met with such a task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)
The resulting identities are useful to remember and apply without intermediate calculations. Short verbal formulations help this.
\((a + b)^2 = a^2 + b^2 + 2ab \) - the square of the sum is equal to the sum of the squares and the double product.
\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is the sum of the squares without doubling the product.
\(a^2 - b^2 = (a - b)(a + b) \) - the difference of squares is equal to the product of the difference and the sum.
These three identities allow in transformations to replace their left parts with right ones and vice versa - right parts with left ones. The most difficult thing in this case is to see the corresponding expressions and understand what the variables a and b are replaced in them. Let's look at a few examples of using abbreviated multiplication formulas.
That part of the equation is the expression in brackets. To open parentheses, look at the sign in front of the parentheses. If there is a plus sign, nothing will change when expanding the brackets in the expression record: just remove the brackets. If there is a minus sign, when opening the brackets, it is necessary to change all the signs that are initially in brackets to the opposite ones. For example, -(2x-3)=-2x+3.
Multiplying two brackets.
If the equation contains the product of two parentheses, expand the parentheses according to the standard rule. Each term of the first parenthesis is multiplied with each term of the second parenthesis. The resulting numbers are summed up. In this case, the product of two "pluses" or two "minuses" gives the term a "plus" sign, and if the factors have different signs, then it gets a minus sign.
Consider .
(5x+1)(3x-4)=5x*3x-5x*4+1*3x-1*4=15x^2-20x+3x-4=15x^2-17x-4.
By expanding parentheses, sometimes raising an expression to . The formulas for squaring and cubeping must be known by heart and remembered.
(a+b)^2=a^2+2ab+b^2
(a-b)^2=a^2-2ab+b^2
(a+b)^3=a^3+3a^2*b+3ab^2+b^3
(a-b)^3=a^3-3a^2*b+3ab^2-b^3
Formulas for raising an expression greater than three can be done using Pascal's triangle.
Sources:
- parenthesis opening formula
Mathematical operations enclosed in brackets can contain variables and expressions of varying degrees of complexity. To multiply such expressions, one will have to look for a solution in general view, expanding the brackets and simplifying the result. If the brackets contain operations without variables, only with numerical values, then it is not necessary to open the brackets, since if a computer is available to its user, very significant computing resources are available - it is easier to use them than to simplify the expression.
Instruction
Multiply successively each (or reduced from) contained in one parenthesis by the contents of all other parentheses if you want to get a general result. For example, let the original expression be written like this: (5+x)∗(6-x)∗(x+2). Then successive multiplication (that is, expanding the brackets) will give the following result: (5+x)∗(6-x)∗(x+2) = (5∗6-5∗x)∗(5∗x+5∗2) + (6∗x-x∗x)∗(x∗x+2∗x) = (5∗6∗5∗x+5∗6∗5∗2) - (5∗x∗5∗x+5∗ x∗5∗2) + (6∗x∗x∗x+6∗x∗2∗x) - (x∗x∗x∗x+x∗x∗2∗x) = 5∗6∗5∗x + 5∗6∗5∗2 - 5∗x∗5∗x - 5∗x∗5∗2 + 6∗x∗x∗x + 6∗x∗2∗x - x∗x∗x∗x - x ∗x∗2∗x = 150∗x + 300 - 25∗x² - 50∗x + 6∗x³ + 12∗x² - x∗x³ - 2∗x³.
Simplify after the result by shortening expressions. For example, the expression obtained in the previous step can be simplified as follows: 150∗x + 300 - 25∗x² - 50∗x + 6∗x³ + 12∗x² - x∗x³ - 2∗x³ = 100∗x + 300 - 13∗ x² - 8∗x³ - x∗x³.
Use a calculator if you need to multiply x equals 4.75, that is, (5+4.75)∗(6-4.75)∗(4.75+2). To calculate this value, go to the Google or Nigma search engine website and enter the expression in the query field in its original form (5+4.75)*(6-4.75)*(4.75+2). Google will show 82.265625 immediately without pressing a button, while Nigma needs to send the data to the server with a button press.
In this lesson, you will learn how to transform an expression that contains parentheses into an expression that does not contain parentheses. You will learn how to open brackets preceded by a plus sign and a minus sign. We will remember how to open brackets using the distributive law of multiplication. The considered examples will allow linking new and previously studied material into a single whole.
Topic: Equation Solving
Lesson: Parentheses expansion
How to open brackets preceded by a "+" sign. Use of the associative law of addition.
If you need to add the sum of two numbers to a number, then you can add the first term to this number, and then the second.
To the left of the equal sign is an expression with parentheses, and to the right is an expression without parentheses. This means that when passing from the left side of the equality to the right side, the brackets were opened.
Consider examples.
Example 1 
Expanding the brackets, we changed the order of operations. Counting has become more convenient.
Example 2 
Example 3
Note that in all three examples, we simply removed the parentheses. Let's formulate the rule:
Comment.
If the first term in brackets is unsigned, then it must be written with a plus sign.
You can follow the step by step example. First, add 445 to 889. This mental action can be performed, but it is not very easy. Let's open the brackets and see that the changed order of operations will greatly simplify the calculations.
If you follow the indicated order of actions, then you must first subtract 345 from 512, and then add 1345 to the result. By expanding the brackets, we will change the order of actions and greatly simplify the calculations.
Illustrative example and rule.
Consider an example: . You can find the value of the expression by adding 2 and 5, and then taking the resulting number with the opposite sign. We get -7.
On the other hand, the same result can be obtained by adding the opposite numbers.
Let's formulate the rule:
Example 1 
Example 2
The rule does not change if there are not two, but three or more terms in brackets.
Example 3 
Comment. Signs are reversed only in front of the terms.
To open parentheses, this case remember the distributive property.
First, multiply the first bracket by 2 and the second by 3.
The first bracket is preceded by a “+” sign, which means that the signs must be left unchanged. The second is preceded by a “-” sign, therefore, all signs must be reversed
Bibliography
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6 - ZSH MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - ZSH MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Interlocutor textbook for grades 5-6 high school. Library of the teacher of mathematics. - Enlightenment, 1989.
- Online math tests ().
- You can download the ones specified in clause 1.2. books().
Homework
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M .: Mnemosyne, 2012. (see link 1.2)
- Homework: No. 1254, No. 1255, No. 1256 (b, d)
- Other assignments: No. 1258(c), No. 1248
In this article, we will consider in detail the basic rules for such an important topic in a mathematics course as opening 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 brackets 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.
