Expanding parentheses a. Opening brackets: rules and examples (grade 7)
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 terms of the polynomial. Monomials are also classified as polynomials, considering a monomial to be a polynomial consisting of one member.
For example, a polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.
Let us represent all terms in the form of monomials of the standard form:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16\)
Let us present 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 terms 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 degree of polynomial of a standard form take the highest of the powers of its members. Thus, the binomial \(12a^2b - 7b\) has the third degree, and the trinomial \(2b^2 -7b + 6\) has the second.
Typically, the terms of standard form polynomials containing one variable are arranged in descending order of exponents. For example:
\(5x - 18x^3 + 1 + x^5 = x^5 - 18x^3 + 5x + 1\)
The sum of several polynomials can be transformed (simplified) into a polynomial of standard form.
Sometimes the terms of a polynomial need to be divided into groups, enclosing each group in parentheses. Since enclosing parentheses is the inverse transformation of opening parentheses, it is easy to formulate rules for opening brackets:
If a “+” sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.
If a “-” sign is placed before the brackets, then the terms enclosed in the brackets are written with opposite signs.
Transformation (simplification) of the product of a monomial and a polynomial
Using the distributive property of multiplication, you 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, you must multiply that monomial by each of the terms of the polynomial.
We have already used this rule several times to multiply by a sum.
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 the following rule is used.
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 squares, differences and difference of squares
You have to deal with some expressions in algebraic transformations more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), i.e. the square of the sum, the square of the difference and difference of squares. You noticed that the names of these expressions seem to be incomplete, 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 does not occur very often; as a rule, instead of the letters a and b, it contains various, sometimes quite complex, expressions.
The expressions \((a + b)^2, \; (a - b)^2 \) can be easily converted (simplified) into polynomials of the standard form; in fact, you have already encountered this task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)
It is useful to remember the resulting identities and apply them without intermediate calculations. Brief 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 equal to the sum of squares without the doubled 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 one to replace its left-hand parts with right-hand ones in transformations and vice versa - right-hand parts with left-hand ones. 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 several examples of using abbreviated multiplication formulas.
In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the “Achilles and the Tortoise” aporia. Here's what it sounds like:Let's say Achilles runs ten times faster than the tortoise and is a thousand steps behind it. During the time it takes Achilles to run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred steps, the tortoise crawls another ten steps, and so on. The process will continue ad infinitum, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert... They all considered Zeno's aporia in one way or another. The shock was so strong that " ... discussions continue to this day; the scientific community has not yet been able to come to a common opinion on the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a generally accepted solution to the problem..."[Wikipedia, "Zeno's Aporia". Everyone understands that they are being fooled, but no one understands what the deception consists of.
From a mathematical point of view, Zeno in his aporia clearly demonstrated the transition from quantity to . This transition implies application instead of permanent ones. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed, or it has not been applied to Zeno’s aporia. Applying our usual logic leads us into a trap. We, due to the inertia of thinking, apply constant units of time to the reciprocal value. From a physical point of view, this looks like time slowing down until it stops completely at the moment when Achilles catches up with the turtle. If time stops, Achilles can no longer outrun the tortoise.
If we turn our usual logic around, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of “infinity” in this situation, then it would be correct to say “Achilles will catch up with the turtle infinitely quickly.”
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal units. In Zeno's language it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise will crawl a hundred steps in the same direction. During the next time interval equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein’s statement about the irresistibility of the speed of light is very similar to Zeno’s aporia “Achilles and the Tortoise”. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells about a flying arrow:
A flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time a flying arrow is at rest at different points in space, which, in fact, is motion. Another point needs to be noted here. From one photograph of a car on the road it is impossible to determine either the fact of its movement or the distance to it. To determine whether a car is moving, you need two photographs taken from the same point at different points in time, but you cannot determine the distance from them. To determine the distance to a car, you need two photographs taken from different points in space at one point in time, but from them you cannot determine the fact of movement (of course, you still need additional data for calculations, trigonometry will help you). What I want to draw special attention to is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Wednesday, July 4, 2018
The differences between set and multiset are described very well on Wikipedia. Let's see.
As you can see, “there cannot be two identical elements in a set,” but if there are identical elements in a set, such a set is called a “multiset.” Reasonable beings will never understand such absurd logic. This is the level of talking parrots and trained monkeys, who have no intelligence from the word “completely”. Mathematicians act as ordinary trainers, preaching to us their absurd ideas.
Once upon a time, the engineers who built the bridge were in a boat under the bridge while testing the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase “mind me, I’m in the house,” or rather, “mathematics studies abstract concepts,” there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash register, giving out salaries. So a mathematician comes to us for his money. We count out the entire amount to him and lay it out on our table in different piles, into which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his “mathematical set of salary.” Let us explain to the mathematician that he will receive the remaining bills only when he proves that a set without identical elements is not equal to a set with identical elements. This is where the fun begins.
First of all, the logic of the deputies will work: “This can be applied to others, but not to me!” Then they will begin to reassure us that bills of the same denomination have different bill numbers, which means they cannot be considered the same elements. Okay, let's count salaries in coins - there are no numbers on the coins. Here the mathematician will begin to frantically remember physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms is unique for each coin...
And now I have the most interesting question: where is the line beyond which the elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science is not even close to lying here.
Look here. We select football stadiums with the same field area. The areas of the fields are the same - which means we have a multiset. But if we look at the names of these same stadiums, we get many, because the names are different. As you can see, the same set of elements is both a set and a multiset. Which is correct? And here the mathematician-shaman-sharpist pulls out an ace of trumps from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I'll show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but that’s why they are shamans, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the page "Sum of digits of a number." She doesn't exist. There is no formula in mathematics that can be used to find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics the task sounds like this: “Find the sum of graphic symbols representing any number.” Mathematicians cannot solve this problem, but shamans can do it easily.
Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let us have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number into a graphical number symbol. This is not a mathematical operation.
2. We cut one resulting picture into several pictures containing individual numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic symbols into numbers. This is not a mathematical operation.
4. Add the resulting numbers. Now this is mathematics.
The sum of the digits of the number 12345 is 15. These are the “cutting and sewing courses” taught by shamans that mathematicians use. But that is not all.
From a mathematical point of view, it does not matter in which number system we write a number. So, in different number systems the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With the large number 12345, I don’t want to fool my head, let’s consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We won't look at every step under a microscope; we've already done that. Let's look at the result.
As you can see, in different number systems the sum of the digits of the same number is different. This result has nothing to do with mathematics. It’s the same as if you determined the area of a rectangle in meters and centimeters, you would get completely different results.
Zero looks the same in all number systems and has no sum of digits. This is another argument in favor of the fact that. Question for mathematicians: how is something that is not a number designated in mathematics? What, for mathematicians nothing exists except numbers? I can allow this for shamans, but not for scientists. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement for numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical operation does not depend on the size of the number, the unit of measurement used and on who performs this action.
Oh! Isn't this the women's restroom?
- Young woman! This is a laboratory for the study of the indephilic holiness of souls during their ascension to heaven! Halo on top and arrow up. What other toilet?
Female... The halo on top and the arrow down are male.
If such a work of design art flashes before your eyes several times a day,
Then it’s not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort to see minus four degrees in a pooping person (one picture) (a composition of several pictures: a minus sign, the number four, a designation of degrees). And I don’t think this girl is a fool who doesn’t know physics. She just has a strong stereotype of perceiving graphic images. And mathematicians teach us this all the time. Here's an example.
1A is not “minus four degrees” or “one a”. This is "pooping man" or the number "twenty-six" in hexadecimal notation. Those people who constantly work in this number system automatically perceive a number and a letter as one graphic symbol.
“Opening parentheses” - Mathematics textbook, grade 6 (Vilenkin)
Short description:
In this section you will learn how to expand parentheses in examples. What is it for? Everything is for the same thing as before - to make it easier and simpler for you to count, to make fewer mistakes, and ideally (the dream of your mathematics teacher) in order to solve everything without mistakes.
You already know that parentheses are placed in mathematical notation if two mathematical signs appear in a row, if we want to show the combination of numbers, their regrouping. Expanding parentheses means getting rid of unnecessary characters. For example: (-15)+3=-15+3=-12, 18+(-16)=18-16=2. Do you remember the distributive property of multiplication relative to addition? Indeed, in that example we also got rid of brackets to simplify calculations. The named property of multiplication can also be applied to four, three, five or more terms. For example: 15*(3+8+9+6)=15*3+15*8+15*9+15*6=390. Have you noticed that when you open the brackets, the numbers in them do not change sign if the number in front of the brackets is positive? After all, fifteen is a positive number. And if you solve this example: -15*(3+8+9+6)=-15*3+(-15)*8+(-15)*9+(-15)*6=-45+(- 120)+(-135)+(-90)=-45-120-135-90=-390. We had a negative number minus fifteen in front of the brackets, when we opened the brackets all the numbers began to change their sign to another - the opposite - from plus to minus.
Based on the above examples, two basic rules for opening parentheses can be stated:
1. If you have a positive number in front of the brackets, then after opening the brackets all the signs of the numbers in the brackets do not change, but remain exactly the same as they were.
2. If you have a negative number in front of the brackets, then after opening the brackets the minus sign is no longer written, and the signs of all absolute numbers in the brackets suddenly change to the opposite.
For example: (13+8)+(9-8)=13+8+9-8=22; (13+8)-(9-8)=13+8-9+8=20. Let's complicate our examples a little: (13+8)+2(9-8)=13+8+2*9-2*8=21+18-16=23. You noticed that when opening the second brackets, we multiplied by 2, but the signs remained the same as they were. Here’s an example: (3+8)-2*(9-8)=3+8-2*9+2*8=11-18+16=9, in this example the number two is negative, it’s before the brackets stands with a minus sign, so when opening them, we changed the signs of the numbers to the opposite ones (nine was with a plus, became a minus, eight was with a minus, became a plus).
In this lesson you will learn how to transform an expression containing parentheses into an expression without parentheses. You will learn how to open parentheses preceded by a plus sign and a minus sign. We will remember how to open brackets using the distributive law of multiplication. The considered examples will allow you to connect new and previously studied material into a single whole.
Topic: Solving equations
Lesson: Expanding Parentheses
How to expand parentheses preceded by a “+” sign. Using the associative law of addition.
If you need to add the sum of two numbers to a number, you can first add the first term to this number, and then the second.
To the left of the equal sign is an expression with parentheses, and to the right is an expression without parentheses. This means that when moving from the left side of the equality to the right, the opening of the parentheses occurred.
Let's look at examples.
Example 1. 
By opening the brackets, we changed the order of actions. It has become more convenient to count.
Example 2. 
Example 3.
Note that in all three examples we simply removed the parentheses. Let's formulate a rule:
Comment.
If the first term in brackets is unsigned, then it must be written with a plus sign.
You can follow the example step by step. First, add 445 to 889. This action can be performed mentally, but it is not very easy. Let's open the brackets and see that the changed procedure will significantly simplify the calculations.
If you follow the indicated procedure, you must first subtract 345 from 512, and then add 1345 to the result. By opening the brackets, we will change the procedure and significantly simplify the calculations.
Illustrating example and rule.
Let's look at an example: . You can find the value of an expression by adding 2 and 5, and then taking the resulting number with the opposite sign. We get -7.
On the other hand, the same result can be obtained by adding the opposite numbers of the original ones.
Let's formulate a rule:
Example 1. 
Example 2.
The rule does not change if there are not two, but three or more terms in brackets.
Example 3. 
Comment. The signs are reversed only in front of the terms.
In order to open the brackets, in this case we need to remember the distributive property.
First, multiply the first bracket by 2, and the second by 3.
The first bracket is preceded by a “+” sign, which means that the signs must be left unchanged. The second sign is preceded by a “-” sign, therefore, all signs need to be changed to the opposite
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. Assignments for the mathematics course grades 5-6 - ZSh MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for 6th grade students at the MEPhI correspondence school. - ZSh MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of secondary school. Math teacher's library. - Enlightenment, 1989.
- Online tests in mathematics ().
- You can download those specified in clause 1.2. books().
Homework
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012. (link see 1.2)
- Homework: No. 1254, No. 1255, No. 1256 (b, d)
- Other tasks: No. 1258(c), No. 1248
Expanding parentheses is a type of expression transformation. In this section we will describe the rules for opening parentheses, and also look at the most common examples of problems.
Yandex.RTB R-A-339285-1
What is opening parentheses?
Parentheses are used to indicate the order in which actions are performed in numeric, literal, and variable expressions. It is convenient to move from an expression with brackets to an identically equal expression without brackets. For example, replace the expression 2 · (3 + 4) with an expression of the form 2 3 + 2 4 without parentheses. This technique is called opening brackets.
Definition 1
Expanding parentheses refers to techniques for getting rid of parentheses and is usually considered in relation to expressions that may contain:
- signs “+” or “-” before parentheses containing sums or differences;
- the product of a number, letter or several letters and the sum or difference, which is placed in brackets.
This is how we are used to viewing the process of opening brackets in the school curriculum. However, no one is stopping us from looking at this action more broadly. We can call parenthesis opening the transition from an expression that contains negative numbers in parentheses to an expression that does not have parentheses. For example, we can go from 5 + (− 3) − (− 7) to 5 − 3 + 7. In fact, this is also an opening of parentheses.
In the same way, we can replace the product of expressions in brackets of the form (a + b) · (c + d) with the sum a · c + a · d + b · c + b · d. This technique also does not contradict the meaning of opening parentheses.
Here's another example. We can assume that any expressions can be used in expressions instead of numbers and variables. For example, the expression x 2 · 1 a - x + sin (b) will correspond to an expression without brackets of the form x 2 · 1 a - x 2 · x + x 2 · sin (b).
One more point deserves special attention, which concerns the peculiarities of recording decisions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as an equality. For example, after expanding the parentheses instead of the expression 3 − (5 − 7) we get the expression 3 − 5 + 7 . We can write both of these expressions as the equality 3 − (5 − 7) = 3 − 5 + 7.
Carrying out actions with cumbersome expressions may require recording intermediate results. Then the solution will have the form of a chain of equalities. For example, 5 − (3 − (2 − 1)) = 5 − (3 − 2 + 1) = 5 − 3 + 2 − 1 or 5 − (3 − (2 − 1)) = 5 − 3 + (2 − 1) = 5 − 3 + 2 − 1 .
Rules for opening parentheses, examples
Let's start looking at the rules for opening parentheses.
For single numbers in brackets
Negative numbers in parentheses are often found in expressions. For example, (− 4) and 3 + (− 4) . Positive numbers in brackets also have a place.
Let us formulate a rule for opening parentheses containing single positive numbers. Let's assume that a is any positive number. Then we can replace (a) with a, + (a) with + a, - (a) with – a. If instead of a we take a specific number, then according to the rule: the number (5) will be written as 5 , expression 3 + (5) without brackets will take the form 3 + 5 , since + (5) is replaced by + 5 , and the expression 3 + (− 5) is equivalent to the expression 3 − 5 , because + (− 5) is replaced by − 5 .
Positive numbers are usually written without using parentheses, since parentheses are unnecessary in this case.
Now consider the rule for opening parentheses that contain a single negative number. + (− a) we replace with − a, − (− a) is replaced by + a. If the expression starts with a negative number (−a), which is written in brackets, then the brackets are omitted and instead (−a) remains − a.
Here are some examples: (− 5) can be written as − 5, (− 3) + 0, 5 becomes − 3 + 0, 5, 4 + (− 3) becomes 4 − 3 , and − (− 4) − (− 3) after opening the brackets takes the form 4 + 3, since − (− 4) and − (− 3) is replaced by + 4 and + 3 .
It should be understood that the expression 3 · (− 5) cannot be written as 3 · − 5. This will be discussed in the following paragraphs.
Let's see what the rules for opening parentheses are based on.
According to the rule, the difference a − b is equal to a + (− b) . Based on the properties of actions with numbers, we can create a chain of equalities (a + (− b)) + b = a + ((− b) + b) = a + 0 = a which will be fair. This chain of equalities, by virtue of the meaning of subtraction, proves that the expression a + (− b) is the difference a − b.
Based on the properties of opposite numbers and the rules for subtracting negative numbers, we can state that − (− a) = a, a − (− b) = a + b.
There are expressions that are made up of a number, minus signs and several pairs of parentheses. Using the above rules allows you to sequentially get rid of brackets, moving from inner to outer brackets or in the opposite direction. An example of such an expression would be − (− ((− (5)))) . Let's open the brackets, moving from inside to outside: − (− ((− (5)))) = − (− ((− 5))) = − (− (− 5)) = − (5) = − 5 . This example can also be analyzed in the opposite direction: − (− ((− (5)))) = ((− (5))) = (− (5)) = − (5) = − 5 .
Under a and b can be understood not only as numbers, but also as arbitrary numeric or alphabetic expressions with a "+" sign in front that are not sums or differences. In all these cases, you can apply the rules in the same way as we did for single numbers in parentheses.
For example, after opening the parentheses the expression − (− 2 x) − (x 2) + (− 1 x) − (2 x y 2: z) will take the form 2 · x − x 2 − 1 x − 2 · x · y 2: z . How did we do it? We know that − (− 2 x) is + 2 x, and since this expression comes first, then + 2 x can be written as 2 x, − (x 2) = − x 2, + (− 1 x) = − 1 x and − (2 x y 2: z) = − 2 x y 2: z.
In products of two numbers
Let's start with the rule for opening parentheses in the product of two numbers.
Let's pretend that a and b are two positive numbers. In this case, the product of two negative numbers − a and − b of the form (− a) · (− b) we can replace with (a · b) , and the products of two numbers with opposite signs of the form (− a) · b and a · (− b) can be replaced with (− a b). Multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus gives a minus.
The correctness of the first part of the written rule is confirmed by the rule for multiplying negative numbers. To confirm the second part of the rule, we can use the rules for multiplying numbers with different signs.
Let's look at a few examples.
Example 1
Let's consider an algorithm for opening parentheses in the product of two negative numbers - 4 3 5 and - 2, of the form (- 2) · - 4 3 5. To do this, replace the original expression with 2 · 4 3 5 . Let's open the brackets and get 2 · 4 3 5 .
And if we take the quotient of negative numbers (− 4) : (− 2), then the entry after opening the brackets will look like 4: 2
In place of negative numbers − a and − b can be any expressions with a minus sign in front that are not sums or differences. For example, these can be products, quotients, fractions, powers, roots, logarithms, trigonometric functions, etc.
Let's open the brackets in the expression - 3 · x x 2 + 1 · x · (- ln 5) . According to the rule, we can make the following transformations: - 3 x x 2 + 1 x (- ln 5) = - 3 x x 2 + 1 x ln 5 = 3 x x 2 + 1 x ln 5.
Expression (− 3) 2 can be converted into the expression (− 3 2) . After this you can expand the brackets: − 3 2.
2 3 · - 4 5 = - 2 3 · 4 5 = - 2 3 · 4 5
Dividing numbers with different signs may also require preliminary expansion of parentheses: (− 5) : 2 = (− 5: 2) = − 5: 2 and 2 3 4: (- 3, 5) = - 2 3 4: 3, 5 = - 2 3 4: 3, 5.
The rule can be used to perform multiplication and division of expressions with different signs. Let's give two examples.
1 x + 1: x - 3 = - 1 x + 1: x - 3 = - 1 x + 1: x - 3
sin (x) (- x 2) = (- sin (x) x 2) = - sin (x) x 2
In products of three or more numbers
Let's move on to products and quotients, which contain a larger number of numbers. To open brackets, the following rule will apply here. If there are an even number of negative numbers, you can omit the parentheses and replace the numbers with their opposites. After this, you need to enclose the resulting expression in new brackets. If there is an odd number of negative numbers, omit the parentheses and replace the numbers with their opposites. After this, the resulting expression must be placed in new brackets and a minus sign must be placed in front of it.
Example 2
For example, take the expression 5 · (− 3) · (− 2) , which is the product of three numbers. There are two negative numbers, therefore we can write the expression as (5 · 3 · 2) and then finally open the brackets, obtaining the expression 5 · 3 · 2.
In the product (− 2, 5) · (− 3) : (− 2) · 4: (− 1, 25) : (− 1) five numbers are negative. therefore (− 2, 5) · (− 3) : (− 2) · 4: (− 1, 25) : (− 1) = (− 2, 5 · 3: 2 · 4: 1, 25: 1) . Having finally opened the brackets, we get −2.5 3:2 4:1.25:1.
The above rule can be justified as follows. Firstly, we can rewrite such expressions as a product, replacing division by multiplication by the reciprocal number. We represent each negative number as the product of a multiplying number and - 1 or - 1 is replaced by (− 1) a.
Using the commutative property of multiplication, we swap factors and transfer all factors equal to − 1 , to the beginning of the expression. The product of an even number minus one is equal to 1, and the product of an odd number is equal to − 1 , which allows us to use the minus sign.
If we did not use the rule, then the chain of actions to open the parentheses in the expression - 2 3: (- 2) · 4: - 6 7 would look like this:
2 3: (- 2) 4: - 6 7 = - 2 3 - 1 2 4 - 7 6 = = (- 1) 2 3 (- 1) 1 2 4 (- 1 ) · 7 6 = = (- 1) · (- 1) · (- 1) · 2 3 · 1 2 · 4 · 7 6 = (- 1) · 2 3 · 1 2 · 4 · 7 6 = = - 2 3 1 2 4 7 6
The above rule can be used when opening parentheses in expressions that represent products and quotients with a minus sign that are not sums or differences. Let's take for example the expression
x 2 · (- x) : (- 1 x) · x - 3: 2 .
It can be reduced to the expression without parentheses x 2 · x: 1 x · x - 3: 2.
Expanding parentheses preceded by a + sign
Consider a rule that can be applied to expand parentheses that are preceded by a plus sign, and the “contents” of those parentheses are not multiplied or divided by any number or expression.
According to the rule, the brackets, together with the sign in front of them, are omitted, while the signs of all terms in the brackets are preserved. If there is no sign before the first term in parentheses, then you need to put a plus sign.
Example 3
For example, we give the expression (12 − 3 , 5) − 7 . By omitting the parentheses, we keep the signs of the terms in parentheses and put a plus sign in front of the first term. The entry will look like (12 − 3, 5) − 7 = + 12 − 3, 5 − 7. In the example given, it is not necessary to place a sign in front of the first term, since + 12 − 3, 5 − 7 = 12 − 3, 5 − 7.
Example 4
Let's look at another example. Let's take the expression x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x and carry out the actions with it x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x = = x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x
Here's another example of expanding parentheses:
Example 5
2 + x 2 + 1 x - x y z + 2 x - 1 + (- 1 + x - x 2) = = 2 + x 2 + 1 x - x y z + 2 x - 1 - 1 + x + x 2
How are parentheses preceded by a minus sign expanded?
Let's consider cases where there is a minus sign in front of the parentheses, and which are not multiplied (or divided) by any number or expression. According to the rule for opening brackets preceded by a “-” sign, brackets with a “-” sign are omitted, and the signs of all terms inside the brackets are reversed.
Example 6
Eg:
1 2 = 1 2 , - 1 x + 1 = - 1 x + 1 , - (- x 2) = x 2
Expressions with variables can be converted using the same rule:
X + x 3 - 3 - - 2 x 2 + 3 x 3 x + 1 x - 1 - x + 2,
we get x - x 3 - 3 + 2 · x 2 - 3 · x 3 · x + 1 x - 1 - x + 2 .
Opening parentheses when multiplying a number by a parenthesis, expressions by a parenthesis
Here we will look at cases where you need to expand parentheses that are multiplied or divided by some number or expression. Formulas of the form (a 1 ± a 2 ± … ± a n) b = (a 1 b ± a 2 b ± … ± a n b) or b · (a 1 ± a 2 ± … ± a n) = (b · a 1 ± b · a 2 ± … ± b · a n), Where a 1 , a 2 , … , a n and b are some numbers or expressions.
Example 7
For example, let's expand the parentheses in the expression (3 − 7) 2. According to the rule, we can carry out the following transformations: (3 − 7) · 2 = (3 · 2 − 7 · 2) . We get 3 · 2 − 7 · 2 .
Opening the parentheses in the expression 3 x 2 1 - x + 1 x + 2, we get 3 x 2 1 - 3 x 2 x + 3 x 2 1 x + 2.
Multiplying parenthesis by parenthesis
Consider the product of two brackets of the form (a 1 + a 2) · (b 1 + b 2) . This will help us obtain a rule for opening parentheses when performing bracket-by-bracket multiplication.
In order to solve the given example, we denote the expression (b 1 + b 2) like b. This will allow us to use the rule for multiplying a parenthesis by an expression. We get (a 1 + a 2) · (b 1 + b 2) = (a 1 + a 2) · b = (a 1 · b + a 2 · b) = a 1 · b + a 2 · b. By performing a reverse replacement b by (b 1 + b 2), again apply the rule of multiplying an expression by a bracket: a 1 b + a 2 b = = a 1 (b 1 + b 2) + a 2 (b 1 + b 2) = = (a 1 b 1 + a 1 b 2) + (a 2 b 1 + a 2 b 2) = = a 1 b 1 + a 1 b 2 + a 2 b 1 + a 2 b 2
Thanks to a number of simple techniques, we can arrive at the sum of the products of each of the terms from the first bracket by each of the terms from the second bracket. The rule can be extended to any number of terms inside the brackets.
Let us formulate the rules for multiplying brackets by brackets: to multiply two sums together, you need to multiply each of the terms of the first sum by each of the terms of the second sum and add the results.
The formula will look like:
(a 1 + a 2 + . . . + a m) · (b 1 + b 2 + . . . + b n) = = a 1 b 1 + a 1 b 2 + . . . + a 1 b n + + a 2 b 1 + a 2 b 2 + . . . + a 2 b n + + . . . + + a m b 1 + a m b 1 + . . . a m b n
Let's expand the brackets in the expression (1 + x) · (x 2 + x + 6) It is the product of two sums. Let's write the solution: (1 + x) · (x 2 + x + 6) = = (1 · x 2 + 1 · x + 1 · 6 + x · x 2 + x · x + x · 6) = = 1 · x 2 + 1 x + 1 6 + x x 2 + x x + x 6
It is worth mentioning separately those cases where there is a minus sign in parentheses along with plus signs. For example, take the expression (1 − x) · (3 · x · y − 2 · x · y 3) .
First, let's present the expressions in brackets as sums: (1 + (− x)) · (3 · x · y + (− 2 · x · y 3)). Now we can apply the rule: (1 + (− x)) · (3 · x · y + (− 2 · x · y 3)) = = (1 · 3 · x · y + 1 · (− 2 · x · y 3) + (− x) · 3 · x · y + (− x) · (− 2 · x · y 3))
Let's open the brackets: 1 · 3 · x · y − 1 · 2 · x · y 3 − x · 3 · x · y + x · 2 · x · y 3 .
Expanding parentheses in products of multiple parentheses and expressions
If there are three or more expressions in parentheses in an expression, the parentheses must be opened sequentially. You need to start the transformation by putting the first two factors in brackets. Within these brackets we can carry out transformations according to the rules discussed above. For example, the parentheses in the expression (2 + 4) · 3 · (5 + 7 · 8) .
The expression contains three factors at once (2 + 4) , 3 and (5 + 7 8) . We will open the brackets sequentially. Let's enclose the first two factors in another bracket, which we'll make red for clarity: (2 + 4) 3 (5 + 7 8) = ((2 + 4) 3) (5 + 7 8).
In accordance with the rule for multiplying a bracket by a number, we can carry out the following actions: ((2 + 4) · 3) · (5 + 7 · 8) = (2 · 3 + 4 · 3) · (5 + 7 · 8) .
Multiply bracket by bracket: (2 3 + 4 3) (5 + 7 8) = 2 3 5 + 2 3 7 8 + 4 3 5 + 4 3 7 8 .
Bracket in kind
Degrees, the bases of which are some expressions written in brackets, with natural exponents can be considered as the product of several brackets. Moreover, according to the rules from the two previous paragraphs, they can be written without these brackets.
Consider the process of transforming the expression (a + b + c) 2 . It can be written as the product of two brackets (a + b + c) · (a + b + c). Let's multiply bracket by bracket and get a · a + a · b + a · c + b · a + b · b + b · c + c · a + c · b + c · c.
Let's look at another example:
Example 8
1 x + 2 3 = 1 x + 2 1 x + 2 1 x + 2 = = 1 x 1 x + 1 x 2 + 2 1 x + 2 2 1 x + 2 = = 1 x · 1 x · 1 x + 1 x · 2 · 1 x + 2 · 1 x · 1 x + 2 · 2 · 1 x + 1 x · 1 x · 2 + + 1 x 2 · 2 + 2 · 1 x · 2 + 2 2 2
Dividing parenthesis by number and parentheses by parenthesis
Dividing a bracket by a number requires that all terms enclosed in brackets be divided by the number. For example, (x 2 - x) : 4 = x 2: 4 - x: 4 .
Division can first be replaced by multiplication, after which you can use the appropriate rule for opening parentheses in a product. The same rule applies when dividing a parenthesis by a parenthesis.
For example, we need to open the parentheses in the expression (x + 2) : 2 3 . To do this, first replace division by multiplying by the reciprocal number (x + 2): 2 3 = (x + 2) · 2 3. Multiply the bracket by the number (x + 2) · 2 3 = x · 2 3 + 2 · 2 3 .
Here's another example of division by parenthesis:
Example 9
1 x + x + 1: (x + 2) .
Let's replace division with multiplication: 1 x + x + 1 · 1 x + 2.
Let's do the multiplication: 1 x + x + 1 · 1 x + 2 = 1 x · 1 x + 2 + x · 1 x + 2 + 1 · 1 x + 2 .
Order of opening brackets
Now let’s consider the order of application of the rules discussed above in general expressions, i.e. in expressions that contain sums with differences, products with quotients, parentheses to the natural degree.
Procedure:
- the first step is to raise the brackets to a natural power;
- at the second stage, the opening of brackets in works and quotients is carried out;
- The final step is to open the parentheses in the sums and differences.
Let's consider the order of actions using the example of the expression (− 5) + 3 · (− 2) : (− 4) − 6 · (− 7) . Let us transform from the expressions 3 · (− 2) : (− 4) and 6 · (− 7) , which should take the form (3 2:4) and (− 6 · 7) . When substituting the obtained results into the original expression, we obtain: (− 5) + 3 · (− 2) : (− 4) − 6 · (− 7) = (− 5) + (3 · 2: 4) − (− 6 · 7) . Open the brackets: − 5 + 3 · 2: 4 + 6 · 7.
When dealing with expressions that contain parentheses within parentheses, it is convenient to carry out transformations by working from the inside out.
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