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 as monomials of the standard form:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16 \)
We give similar terms in the resulting polynomial:
\(8b^5 -14b^5 +3b^2 -8b -3b^2 + 16 = -6b^5 -8b + 16 \)
The result is a polynomial, all members of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.
Per 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 noticed that the names of the indicated 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. Concise 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.
Parentheses are used to indicate the order in which actions are performed in numeric and alphabetic 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. Both of these expressions can be written as the equality 3−(5−7)=3−5+7.
And one more important point. In mathematics, to reduce entries, it is customary not to write a plus sign if it is the first in an expression or in brackets. For example, if we add two positive numbers, for example, seven and three, then we write not +7 + 3, but simply 7 + 3, despite the fact that seven is also a positive number. Similarly, if you see, for example, the expression (5 + x) - know that there is a plus in front of the bracket, which is not written, and there is a plus + (+5 + x) in front of the five.
Bracket expansion rule for addition
When opening brackets, if there is a plus before the brackets, then this plus is omitted along with the brackets.
Example. Open the brackets in the expression 2 + (7 + 3) Before the brackets plus, then the characters in front of the numbers in the brackets do not change.
2 + (7 + 3) = 2 + 7 + 3
The rule for expanding brackets when subtracting
If there is a minus before the brackets, then this minus is omitted along with the brackets, but the terms that were in the brackets change their sign to the opposite. The absence of a sign before the first term in parentheses implies a + sign.
Example. Open brackets in expression 2 − (7 + 3)
There is a minus before the brackets, so you need to change the signs before the numbers from the brackets. There is no sign in brackets before the number 7, which means that the seven is positive, it is considered that the + sign is in front of it.
2 − (7 + 3) = 2 − (+ 7 + 3)
When opening the brackets, we remove the minus from the example, which was before the brackets, and the brackets themselves 2 − (+ 7 + 3), and change the signs that were in the brackets to the opposite ones.
2 − (+ 7 + 3) = 2 − 7 − 3
Expanding parentheses when multiplying
If there is a multiplication sign in front of the brackets, then each number inside the brackets is multiplied by the factor in front of the brackets. At the same time, multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus, gives a minus.
Thus, parentheses in 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
In almost any text, you can find brackets and dashes. But users do not always draw them correctly. For example, it's not uncommon to see dashes without one or two spaces when text sticks to a character. The same applies to brackets, the use of which is out of place or without taking into account the rules of writing overloads the text. This article discusses the issues of writing brackets and dashes in accordance with generally accepted rules.
Parentheses rules
When writing brackets, follow the same rules as for quotation marks. For example, two parentheses are not placed in a row.
There are several cases where brackets are used:
Separate words, groups of words and whole sentences that are not directly related to the main idea expressed by the author. Phrases uttered in passing, when the author does not draw the attention of the reader to them. Expressions in brackets fall out of the syntactic structure of the sentence.
Example: " And although I myself understand that when she pulls my whirlwinds, she pulls them out only from the pity of her heart (for, I repeat without embarrassment, she pulls my whirlwinds, young man, ”he confirmed with extreme dignity, hearing another giggle) , but, God, what if she even once ... But no! No! All this is in vain, and there is nothing to say! there is nothing to say! .. for more than once the desired has already happened, and more than once they have pitied me, but ... such is already my trait, and I am a born cattle!" (F.M. Dostoevsky, "Crime and Punishment")
Brief remarks to explain a particular word or phrase in a sentence are placed in brackets.
Example: " Went normal, soothing chatter, when, together with sincere sympathy (we all belong here, and all, in general, are kind people) there is also a hint of mocking relief. Not me! I did not do this stupidity, - it was read in the faces."(S. Lukyanenko, "Shadows of Dreams")
Example: " I asked a tipsy yogi
(He razors, he ate nails like sausage):
“Listen, friend, open up to me - by God,
I'll take the secret with me to the grave!»
(V. Vysotsky, "A song about yogis")
References to formulas and illustrations are enclosed in parentheses, for example (fig. 2), (diag. 3, p. 184) , « Formula (1) is a consequence of the Pythagorean theorem. Formulas (2) and (3) are obtained from the formula (1) . » and sources of information (literature, publications) in square brackets, for example: , , etc.
Remarks are enclosed in brackets, a vivid example is scenarios where the verbal embodiment of continuous action is indicated in the remarks, for example:
« Will laughs.
SKYLAR (continues)
How do you do this? I don't... I mean, even the smartest people I know, we have a couple at Harvard, we have to study - a lot. It's complicated.
(pause)
Look, Will, if you don't want to tell me...»
(Script for the film "Good Will Hunting"
Brackets are also used when adding unfinished words in author's papers.
Numbering in the text is written using brackets in the following format:
1)
a)
*)
Signs of footnotes (references) are drawn up in a similar way.
Dash rules
A dash refers to punctuation marks; when writing before and after a dash, a space is always written.
There are a few exceptions when a dash is written without both or one space:
when a paragraph begins with a dash, a space is placed only after.
when a dash stands between two numbers, acting as a hyphen. For example: " every day our site is visited by 3000 -
3500 visitors».
For example: " – Oh-oh… Uh… –
only and was able to mumble dumbfounded Paige.(Philip K. Dick, Minority Report)
Most punctuation marks, including commas, question marks, exclamation marks, are placed before the dash. Example: " The central mountainous region in which the Pindus mountains are located , - the most sparsely populated. The highest point in Greece, Mount Olympus (2917 m) is located in this region. Central Greece is the most populated region."(Eklopedic reference book" The whole world. Countries ")
The dash is used in several ways:
- as a punctuation mark;
- as a connector of a pair of limit numbers, for example: 80-90%
;
- as a mathematical minus sign;
- as a separator symbol or symbol from the explanatory text, for example, when a decoding of the symbols included in the formula is given, or an explanation is given for the illustration;
- as a hyphen, with the dash written together with the non-portable part of the word and should not be repeated at the beginning of the next line;
- as a connecting dash or hyphen.
The main function of brackets 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 \(5·(3+7)\), addition in brackets will be calculated first, and only then multiplication: \(5·(3+7)=5·10=50\).
Example.
Expand the bracket: \(-(4m+3)\).
Solution
: \(-(4m+3)=-4m-3\).
Example.
Expand the bracket and give like terms \(5-(3x+2)+(2+3x)\).
Solution
: \(5-(3x+2)+(2+3x)=5-3x-2+2+3x=5\).
Example.
Expand the brackets \(5(3-x)\).
Solution
: We have \(3\) and \(-x\) in the bracket, and five in front of the bracket. This means that each member of the bracket is multiplied by \ (5 \) - I remind you that the multiplication sign between a number and a bracket in mathematics is not written to reduce the size of records.
Example.
Expand the brackets \(-2(-3x+5)\).
Solution
: As in the previous example, the bracketed \(-3x\) and \(5\) are multiplied by \(-2\).
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 parenthesis by parenthesis, each term of the first parenthesis is multiplied with every 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 opened 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 - each of its members is multiplied by the second bracket:
Step 2. Expand the products of the bracket by the factor as described above:
- the first one first...
Then the second.
Step 3. Now we multiply and bring like terms:
It is not necessary to paint all the transformations in detail, you can immediately multiply. But if you are just learning to open brackets - write in detail, there will be less chance of making a mistake.
Note to the entire section. In fact, you don't need to remember all four rules, you only need to remember one, this one: \(c(a-b)=ca-cb\) . Why? Because if we substitute one instead of c, we get the rule \((a-b)=a-b\) . And if we substitute minus one, we get the rule \(-(a-b)=-a+b\) . Well, if you substitute another bracket instead of c, you can get the last rule.
parenthesis within parenthesis
Sometimes in practice there are problems with brackets nested inside other brackets. Here is an example of such a task: to simplify the expression \(7x+2(5-(3x+y))\).
To be successful in these tasks, you need to:
- carefully understand the nesting of brackets - which one is in which;
- open the brackets sequentially, starting, for example, with the innermost one.
It is important when opening one of the brackets don't touch the rest of the expression, just rewriting it as is.
Let's take the task above as an example.
Example.
Expand the brackets and give like terms \(7x+2(5-(3x+y))\).
Solution:
Example.
Expand the brackets and give like terms \(-(x+3(2x-1+(x-5)))\).
Solution
:
|
\(-(x+3(2x-1\)\(+(x-5)\) \())\) |
This is a triple nesting of parentheses. We start with the innermost one (highlighted in green). There is a plus in front of the parenthesis, so it is simply removed. |
|
|
\(-(x+3(2x-1\)\(+x-5\) \())\) |
Now you need to open the second bracket, intermediate. But before that, we will simplify the expression by ghosting similar terms in this second bracket. |
|
|
\(=-(x\)\(+3(3x-6)\) \()=\) |
Now we open the second bracket (highlighted in blue). There is a multiplier in front of the parenthesis - so each term in the parenthesis is multiplied by it. |
|
|
\(=-(x\)\(+9x-18\) \()=\) |
||
|
And open the last parenthesis. Before the bracket minus - so all the signs are reversed. |
||
Bracket opening is a basic skill in mathematics. Without this skill, it is impossible to have a grade above three in grades 8 and 9. Therefore, I recommend that you understand this topic well.
A + (b + c) can be written without brackets: a + (b + c) \u003d a + b + c. This operation is called parenthesis expansion.
Example 1 Let's open the brackets in the expression a + (- b + c).
Solution. a + (-b + c) \u003d a + ((-b) + c) \u003d a + (-b) + c \u003d a-b + c.
If there is a “+” sign before the brackets, then you can omit the brackets and this “+” sign, retaining the signs of the terms in brackets. If the first term in brackets is written without a sign, then it must be written with a “+” sign.
Example 2 Let's find the value of the expression -2.87+ (2.87-7.639).
Solution. Opening the brackets, we get - 2.87 + (2.87 - 7.639) \u003d - - 2.87 + 2.87 - 7.639 \u003d 0 - 7.639 \u003d - 7.639.
To find the value of the expression - (- 9 + 5), you need to add numbers-9 and 5 and find the number opposite to the amount received: -(- 9 + 5)= -(- 4) = 4.
The same value can be obtained in a different way: first write down the numbers opposite to these terms (i.e. change their signs), and then add: 9 + (- 5) = 4. Thus, - (- 9 + 5) = 9 - 5 = 4.
To write the sum opposite to the sum of several terms, it is necessary to change the signs of these terms.
So - (a + b) \u003d - a - b.
Example 3 Find the value of the expression 16 - (10 -18 + 12).
Solution. 16-(10 -18 + 12) = 16 + (-(10 -18 + 12)) = = 16 + (-10 +18-12) = 16-10 +18-12 = 12.
To open the brackets preceded by the “-” sign, you need to replace this sign with “+”, changing the signs of all the terms in the brackets to the opposite ones, and then open the brackets.
Example 4 Let's find the value of the expression 9.36-(9.36 - 5.48).
Solution. 9.36 - (9.36 - 5.48) = 9.36 + (- 9.36 + 5.48) == 9.36 - 9.36 + 5.48 = 0 -f 5.48 = 5 .48.
Bracket opening and the use of commutative and associative properties additions make calculations easier.
Example 5 Find the value of the expression (-4-20)+(6+13)-(7-8)-5.
Solution. First, we open the brackets, and then we find separately the sum of all positive and separately the sum of all negative numbers, and, finally, add the results:
(- 4 - 20)+(6+ 13)-(7 - 8) - 5 = -4-20 + 6 + 13-7 + 8-5 = = (6 + 13 + 8)+(- 4 - 20 - 7 - 5)= 27-36=-9.
Example 6 Let's find the value of the expression
Solution. First, we represent each term as the sum of their integer and fractional parts, then open the brackets, then add the whole and separately fractional parts and finally sum up the results:
How do you open parentheses that are preceded by a "+" sign? How can you find the value of an expression that is the opposite of the sum of several numbers? How to open brackets preceded by a "-" sign?
1218. Expand the brackets:
a) 3.4+(2.6+ 8.3); c) m+(n-k);
b) 4.57+(2.6 - 4.57); d) c+(-a + b).
1219. Find the value of the expression:
1220. Expand the brackets:
a) 85+(7.8+ 98); d) -(80-16) + 84; g) a-(b-k-n);
b) (4.7 -17) + 7.5; e) -a + (m-2.6); h) - (a-b + c);
c) 64-(90 + 100); e) c+(-a-b); i) (m-n)-(p-k).
1221. Expand the brackets and find the value of the expression:
1222. Simplify the expression:
1223. Write amount two expressions and simplify it:
a) - 4 - m and m + 6.4; d) a + b and p - b
b) 1.1+a and -26-a; e) - m + n and -k - n;
c) a + 13 and -13 + b; e)m - n and n - m.
1224. Write the difference of two expressions and simplify it:
1226. Use the equation to solve the problem:
a) There are 42 books on one shelf, and 34 on the other. Several books were removed from the second shelf, and as many as were left on the second from the first. After that, 12 books remained on the first shelf. How many books were taken off the second shelf?
b) There are 42 students in the first class, 3 students less in the second than in the third. How many students are in the third grade if there are 125 students in these three grades?
1227. Find the value of the expression:
1228. Calculate orally:
1229. Find the largest value of the expression:
1230. Enter 4 consecutive integers if:
a) the smaller of them is equal to -12; c) the smaller of them is equal to n;
b) the greater of them is equal to -18; d) the larger of them is equal to k.
