In this article we will take a detailed look at the basic rules of such an important topic in a mathematics course as opening parentheses. You need to know the rules for opening parentheses in order to correctly solve equations in which they are used.
How to open parentheses correctly 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, the signs inside them do not change when the brackets are opened. Example:
(9 + 3) + (1 - 6 + 9) = 9 + 3 + 1 - 6 + 9 = 16.
How to expand parentheses preceded by a "-" sign
In this case, you need to rewrite all terms without brackets, but at the same time change all the signs inside them to the opposite ones. The signs change only for 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
Before the brackets there is a multiplier number
In this case, you need to multiply each term by a factor and open the brackets without changing the signs. If the multiplier has a “-” sign, then during multiplication 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 parentheses 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 parentheses in a square
If the sum or difference of two terms is squared, the brackets should be opened 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 expand parentheses to another degree
If the sum or difference of terms is raised, for example, to the 3rd or 4th power, then you just need to break the power of the bracket into “squares”. The powers of identical factors are added, and when dividing, the power of the divisor is subtracted from the power 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 together, and then multiply the sum of this multiplication by the terms of the third bracket. Example:
(1 + 2) * (3 + 4) * (5 - 6) = (3 + 4 + 6 + 8) * (5 - 6) = - 21.
These rules for opening parentheses apply equally to solving both linear and trigonometric equations.
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.
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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 a 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 instead of numbers and variables in expressions. For example, the expression x 2 · 1 a - x + sin (b) will correspond to an expression without parentheses 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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In the previous lesson we dealt with factorization. We mastered two methods: putting the common factor out of brackets and grouping. In this lesson - the following powerful method: abbreviated multiplication formulas. In short - FSU.
Abbreviated multiplication formulas (sum and difference square, sum and difference cube, difference of squares, sum and difference of cubes) are extremely necessary in all branches of mathematics. They are used in simplifying expressions, solving equations, multiplying polynomials, reducing fractions, solving integrals, etc. and so on. In short, there is every reason to deal with them. Understand where they come from, why they are needed, how to remember them and how to apply them.
Do we understand?)
Where do abbreviated multiplication formulas come from?
Equalities 6 and 7 are not written in a very familiar way. It's kind of the opposite. This is on purpose.) Any equality works both from left to right and from right to left. This entry makes it clearer where the FSUs come from.
They are taken from multiplication.) For example:
(a+b) 2 =(a+b)(a+b)=a 2 +ab+ba+b 2 =a 2 +2ab+b 2
That's it, no scientific tricks. We simply multiply the brackets and give similar ones. This is how it turns out all abbreviated multiplication formulas. Abbreviated multiplication is because in the formulas themselves there is no multiplication of brackets and reduction of similar ones. Abbreviated.) The result is immediately given.
FSU needs to be known by heart. Without the first three, you can’t dream of a C; without the rest, you can’t dream of a B or A.)
Why do we need abbreviated multiplication formulas?
There are two reasons to learn, even memorize, these formulas. The first is that a ready-made answer automatically reduces the number of errors. But this is not the main reason. But the second one...
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
The main function of parentheses is to change the order of actions when calculating values. For example, in the numerical expression \(5·3+7\) the multiplication will be calculated first, and then the addition: \(5·3+7 =15+7=22\). But in the expression \(5·(3+7)\) the addition in brackets will be calculated first, and only then the multiplication: \(5·(3+7)=5·10=50\).
Example.
Expand the bracket: \(-(4m+3)\).
Solution
: \(-(4m+3)=-4m-3\).
Example.
Open the bracket and give similar terms \(5-(3x+2)+(2+3x)\).
Solution
: \(5-(3x+2)+(2+3x)=5-3x-2+2+3x=5\).
Example.
Expand the brackets \(5(3-x)\).
Solution
: In the bracket we have \(3\) and \(-x\), and before the bracket there is a five. This means that each member of the bracket is multiplied by \(5\) - I remind you that The multiplication sign between a number and a parenthesis is not written in mathematics to reduce the size of entries.
Example.
Expand the brackets \(-2(-3x+5)\).
Solution
: As in the previous example, the \(-3x\) and \(5\) in the parenthesis are multiplied by \(-2\).
Example.
Simplify the expression: \(5(x+y)-2(x-y)\).
Solution
: \(5(x+y)-2(x-y)=5x+5y-2x+2y=3x+7y\).
It remains to consider the last situation.
When multiplying a bracket by a bracket, each term of the first bracket is multiplied with each term of the second:
\((c+d)(a-b)=c·(a-b)+d·(a-b)=ca-cb+da-db\)
Example.
Expand the brackets \((2-x)(3x-1)\).
Solution
: We have a product of brackets and it can be expanded immediately using the formula above. But in order not to get confused, let's do everything step by step.
Step 1. Remove the first bracket - multiply each of its terms by the second bracket:
Step 2. Expand the products of the brackets and the factor as described above:
- First things first...
Then the second.
Step 3. Now we multiply and present similar terms:
It is not necessary to describe all the transformations in such detail; you can multiply them right away. But if you are just learning how to open parentheses, write in detail, there will be less chance of making mistakes.
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 you substitute one instead of c, you 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 a parenthesis
Sometimes in practice there are problems with brackets nested inside other brackets. Here is an example of such a task: simplify the expression \(7x+2(5-(3x+y))\).
To successfully solve such tasks, you need:
- 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 look at the task written above as an example.
Example.
Open the brackets and give similar terms \(7x+2(5-(3x+y))\).
Solution:
Example.
Open the brackets and give similar terms \(-(x+3(2x-1+(x-5)))\).
Solution
:
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\(-(x+3(2x-1\)\(+(x-5)\) \())\) |
There is triple nesting of parentheses here. Let's start with the innermost one (highlighted in green). There is a plus in front of the bracket, so it simply comes off. |
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\(-(x+3(2x-1\)\(+x-5\) \())\) |
Now you need to open the second bracket, the intermediate one. But before that, we will simplify the expression of the ghost-like terms in this second bracket. |
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\(=-(x\)\(+3(3x-6)\) \()=\) |
Now we open the second bracket (highlighted in blue). Before the bracket is a factor - so each term in the bracket is multiplied by it. |
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\(=-(x\)\(+9x-18\) \()=\) |
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And open the last bracket. There is a minus sign in front of the bracket, so all signs are reversed. |
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Expanding parentheses is a basic skill in mathematics. Without this skill, it is impossible to have a grade above a C in 8th and 9th grade. Therefore, I recommend that you understand this topic well.
Let us now consider the squaring of a binomial and, applying an arithmetic point of view, we will speak of the square of the sum, i.e. (a + b)², and the square of the difference of two numbers, i.e. (a – b)².
Since (a + b)² = (a + b) ∙ (a + b),
then we find: (a + b) ∙ (a + b) = a² + ab + ab + b² = a² + 2ab + b², i.e.
(a + b)² = a² + 2ab + b²
It is useful to remember this result both in the form of the above-described equality and in words: the square of the sum of two numbers is equal to the square of the first number plus the product of two by the first number and the second number, plus the square of the second number.
Knowing this result, we can immediately write, for example:
(x + y)² = x² + 2xy + y²
(3ab + 1)² = 9a² b² + 6ab + 1
(x n + 4x)² = x 2n + 8x n+1 + 16x 2
Let's look at the second of these examples. We need to square the sum of two numbers: the first number is 3ab, the second 1. The result should be: 1) the square of the first number, i.e. (3ab)², which is equal to 9a²b²; 2) the product of two by the first number and the second, i.e. 2 ∙ 3ab ∙ 1 = 6ab; 3) the square of the 2nd number, i.e. 1² = 1 - all these three terms must be added together.
We also obtain a formula for squaring the difference of two numbers, i.e. for (a – b)²:
(a – b)² = (a – b) (a – b) = a² – ab – ab + b² = a² – 2ab + b².
(a – b)² = a² – 2ab + b²,
i.e. the square of the difference of two numbers is equal to the square of the first number, minus the product of two by the first number and the second, plus the square of the second number.
Knowing this result, we can immediately perform the squaring of binomials, which, from an arithmetic point of view, represent the difference of two numbers.
(m – n)² = m² – 2mn + n²
(5ab 3 – 3a 2 b) 2 = 25a 2 b 6 – 30a 3 b 4 + 9a 4 b 2 
(a n-1 – a) 2 = a 2n-2 – 2a n + a 2, etc.
Let's explain the 2nd example. Here we have in brackets the difference of two numbers: the first number is 5ab 3 and the second number is 3a 2 b. The result should be: 1) the square of the first number, i.e. (5ab 3) 2 = 25a 2 b 6, 2) the product of two by the 1st and 2nd number, i.e. 2 ∙ 5ab 3 ∙ 3a 2 b = 30a 3 b 4 and 3) the square of the second number, i.e. (3a 2 b) 2 = 9a 4 b 2 ; The first and third terms must be taken with a plus, and the 2nd with a minus, we get 25a 2 b 6 – 30a 3 b 4 + 9a 4 b 2. To explain the 4th example, we only note that 1) (a n-1)2 = a 2n-2 ... the exponent must be multiplied by 2 and 2) the product of two by the 1st number and by the 2nd = 2 ∙ a n-1 ∙ a = 2a n .
If we take the point of view of algebra, then both equalities: 1) (a + b)² = a² + 2ab + b² and 2) (a – b)² = a² – 2ab + b² express the same thing, namely: the square of the binomial is equal to the square of the first term, plus the product of the number (+2) by the first term and the second, plus the square of the second term. This is clear because our equalities can be rewritten as:
1) (a + b)² = (+a)² + (+2) ∙ (+a) (+b) + (+b)²
2) (a – b)² = (+a)² + (+2) ∙ (+a) (–b) + (–b)²
In some cases, it is convenient to interpret the resulting equalities in this way:
(–4a – 3b)² = (–4a)² + (+2) (–4a) (–3b) + (–3b)²
Here we square a binomial whose first term = –4a and second = –3b. Next we get (–4a)² = 16a², (+2) (–4a) (–3b) = +24ab, (–3b)² = 9b² and finally:
(–4a – 3b)² = 6a² + 24ab + 9b²
It would also be possible to obtain and remember the formula for squaring a trinomial, a quadrinomial, or any polynomial in general. However, we will not do this, because we rarely need to use these formulas, and if we need to square any polynomial (except a binomial), we will reduce the matter to multiplication. For example:
31. Let us apply the obtained 3 equalities, namely:
(a + b) (a – b) = a² – b²
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
to arithmetic.
Let it be 41 ∙ 39. Then we can represent this in the form (40 + 1) (40 – 1) and reduce the matter to the first equality - we get 40² – 1 or 1600 – 1 = 1599. Thanks to this, it is easy to perform multiplications like 21 ∙ 19; 22 ∙ 18; 31 ∙ 29; 32 ∙ 28; 71 ∙ 69, etc.
Let it be 41 ∙ 41; it’s the same as 41² or (40 + 1)² = 1600 + 80 + 1 = 1681. Also 35 ∙ 35 = 35² = (30 + 5)² = 900 + 300 + 25 = 1225. If you need 37 ∙ 37, then this is equal to (40 – 3)² = 1600 – 240 + 9 = 1369. Such multiplications (or squaring two-digit numbers) are easy to perform, with some skill, in your head.