Algebraic expression

an expression made up of letters and numbers connected by the signs of addition, subtraction, multiplication, division, raising to an integer power and extracting the root (the exponents and roots must be constant numbers). A.v. is called rational with respect to some letters included in it if it does not contain them under the sign of root extraction, for example

rational with respect to a, b and c. A.v. is called an integer with respect to some letters if it does not contain division into expressions containing these letters, for example 3a/c + bc 2 - 3ac/4 is integer with respect to a and b. If some of the letters (or all) are considered variables, then A.c. is an algebraic function.

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what an “Algebraic expression” is in other dictionaries:

An expression made up of letters and numbers connected by signs algebraic operations: addition, subtraction, multiplication, division, exponentiation, root extraction... Big encyclopedic Dictionary

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An algebraic expression is one or more algebraic quantities (numbers and letters) connected by signs of algebraic operations: addition, subtraction, multiplication and division, as well as taking roots and raising to whole numbers... ... Wikipedia

An expression made up of letters and numbers connected by signs of algebraic operations: addition, subtraction, multiplication, division, exponentiation, root extraction. * * * ALGEBRAIC EXPRESSION ALGEBRAIC EXPRESSION, expression,... ... encyclopedic Dictionary

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Algebra lessons introduce us to different types of expressions. As new material becomes available, expressions become more complex. As you become familiar with degrees, they are gradually added to the expression, complicating it. This also happens with fractions and other expressions.

To make studying the material as convenient as possible, this is done using certain names so that they can be highlighted. This article will give full review all basic school algebraic expressions.

Monomials and polynomials

Expressions monomials and polynomials are studied in school curriculum starting from 7th grade. Definitions of this type were given in textbooks.

Definition 1

Monomials– these are numbers, variables, their powers with a natural exponent, any products made with their help.

Definition 2

Polynomials called the sum of monomials.

If we take, for example, the number 5, the variable x, the degree z 7, then products of the form 5 x And 7 x 2 7 z 7 are considered monomials. When taking the sum of monomials of the form 5+x or z 7 + 7 + 7 x 2 7 z 7, then we get a polynomial.

To distinguish a monomial from a polynomial, pay attention to the degrees and their definitions. The concept of coefficient is important. When reducing similar terms, they are divided by the free term of the polynomial or the leading coefficient.

Most often, some actions are performed on monomials and polynomials, after which the expression is reduced to the form of a monomial. Performs addition, subtraction, multiplication and division, relying on an algorithm to perform operations on polynomials.

When there is one variable, it is possible to divide the polynomial into polynomials, which are represented as a product. This action is called factoring a polynomial.

Rational (algebraic) fractions

The concept of rational fractions is studied in 8th grade high school. Some authors call them algebraic fractions.

Definition 3

Rational algebraic fraction called a fraction in which polynomials or monomials or numbers appear in place of the numerator and denominator.

Let's consider the example of writing rational fractions of the type 3 x + 2, 2 · a + 3 · b 4, x 2 + 1 x 2 - 2 and 2 2 · x + - 5 1 5 · y 3 · x x 2 + 4. Based on the definition, we can say that every fraction is considered a rational fraction.

Algebraic fractions can be added, subtracted, multiplied, divided, and raised to powers. This is discussed in more detail in the section on operations with algebraic fractions. If it is necessary to convert a fraction, they often use the property of reduction and reduction to a common denominator.

Rational Expressions

In the school course, the concept of irrational fractions is studied, since work with rational expressions is necessary.

Definition 4

Rational Expressions are considered numeric and alphabetic expressions where used rational numbers and letters with addition, subtraction, multiplication, division, and raising to a whole power.

Rational expressions may not have signs belonging to the function, which lead to irrationality. Rational expressions do not contain roots, powers with fractional irrational exponents, powers with variables in the exponent, logarithmic expressions, trigonometric functions, and so on.

Based on the rule given above, we will give examples of rational expressions. From the above definition we have that both a numerical expression of the form 1 2 + 3 4 and 5, 2 + (- 0, 1) 2 2 - 3 5 - 4 3 4 + 2: 12 7 - 1 + 7 - 2 2 3 3 - 2 1 + 0, 3 are considered rational. Expressions containing letter designations are also classified as rational a 2 + b 2 3 · a - 0, 5 · b, с variables of the form a x 2 + b x + c and x 2 + x y - y 2 1 2 x - 1 .

All rational expressions are divided into integers and fractions.

Whole rational expressions

Definition 5

Whole rational expressions– these are expressions that do not contain division into expressions with variables of negative degree.

From the definition we have that a whole rational expression is also an expression containing letters, for example, a + 1, an expression containing several variables, for example, x 2 · y 3 − z + 3 2 and a + b 3.

Expressions like x: (y − 1) and 2 x + 1 x 2 - 2 x + 7 - 4 cannot be rational integers, since they have division into an expression with variables.

Fractional rational expressions

Definition 6

Fractional rational expression is an expression that contains division by an expression with variables of negative degree.

From the definition it follows that fractional rational expressions can be 1: x, 5 x 3 - y 3 + x + x 2 and 3 5 7 - a - 1 + a 2 - (a + 1) (a - 2) 2.

If we consider expressions of this type (2 x − x 2) : 4 and a 2 2 - b 3 3 + c 4 + 1 4, 2, then they are not considered fractional rationals, since they do not have expressions with variables in the denominator.

Expressions with powers

Definition 7

Expressions that contain powers in any part of the notation are called expressions with powers or power expressions.

For the concept, we give an example of such an expression. They may not contain variables, for example, 2 3, 32 - 1 5 + 1, 5 3, 5 5 - 2 5 - 1, 5. Power expressions of the form 3 · x 3 · x - 1 + 3 x , x · y 2 1 3 are also typical. In order to solve them, it is necessary to perform some transformations.

Irrational expressions, expressions with roots

The root that occurs in the expression gives it a different name. They are called irrational.

Definition 8

Irrational expressions are expressions that have root signs in their writing.

From the definition it is clear that these are expressions of the form 64, x - 1 4 3 + 3 3, 2 + 1 2 - 1 - 2 + 3 2, a + 1 a 1 2 + 2, x y, 3 x + 1 + 6 x 2 + 5 x and x + 6 + x - 2 3 + 1 4 x 2 3 + 3 - 1 1 3 . Each of them has at least one root icon. Roots and powers are related, so you can see expressions such as x 7 3 - 2 5, n 4 8 · m 3 5: 4 · m 2 n + 3.

Trigonometric expressions

Definition 9

Trigonometric expression- these are expressions containing sin, cos, tg and ctg and their inverses - arcsin, arccos, arctg and arcctg.

Examples of trigonometric functions are obvious: sin π 4 · cos π 6 cos 6 x - 1 and 2 sin x · t g 2 x + 3, 4 3 · t g π - arcsin - 3 5.

To work with such functions, it is necessary to use the properties and basic formulas of direct and inverse functions. The article transformation of trigonometric functions will reveal this issue in more detail.

Logarithmic Expressions

After becoming familiar with logarithms, you can talk about complex logarithmic expressions.

Definition 10

Expressions that have logarithms are called logarithmic.

An example of such functions would be log 3 9 + ln e, log 2 (4 a b), log 7 2 (x 7 3) log 3 2 x - 3 5 + log x 2 + 1 (x 4 + 2) .

You can find expressions where there are powers and logarithms. This is understandable, since from the definition of the logarithm it follows that it is an exponent. Then we get expressions of the form x l g x - 10, log 3 3 x 2 + 2 x - 3, log x + 1 (x 2 + 2 x + 1) 5 x - 2.

To deepen your study of the material, you should refer to the material on converting logarithmic expressions.

Fractions

There are expressions of a special type, which are called fractions. Since they have a numerator and a denominator, they can contain not just numerical values, but also expressions of any type. Let's look at the definition of a fraction.

Definition 11

Fraction is an expression that has a numerator and a denominator, in which there are both numerical and alphabetic designations or expressions.

Examples of fractions that have numbers in the numerator and denominator look like this: 1 4, 2, 2 - 6 2 7, π 2, - e π, (− 15) (− 2) . The numerator and denominator can contain both numerical and alphabetic expressions of the form (a + 1) 3, (a + b + c) (a 2 + b 2) , 1 3 + 1 - 1 3 - 1 1 1 + 1 1 + 1 5, cos 2 α - sin 2 α 1 + 3 t g α, 2 + ln 5 ln x.

Although expressions such as 2 5 − 3 7 , x x 2 + 1: 5 are not fractions, they do have a fraction in their notation.

General expression

Senior grades consider problems of increased difficulty, which contain all the combined tasks of group C on the Unified State Exam. These expressions are particularly complex and contain various combinations of roots, logarithms, powers, and trigonometric functions. These are tasks like x 2 - 1 · sin x + π 3 or sin a r c t g x - a · x 1 + x 2 .

Their appearance suggests that they can be classified as any of the above types. Most often they are not classified as any one, since they have a specific combined solution. They are considered as general expressions, and no additional specifications or expressions are used for the description.

When solving such an algebraic expression, it is always necessary to pay attention to its notation, the presence of fractions, powers or additional expressions. This is necessary in order to accurately determine how to solve it. If you are not sure of its name, then it is recommended to call it an expression of a general type and solve it according to the algorithm written above.

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In algebra lessons at school we come across expressions various types. As you learn new material, recording expressions becomes more diverse and complex. For example, we got acquainted with powers - powers appeared in expressions, we studied fractions - fractional expressions appeared, etc.

For the convenience of describing the material, expressions consisting of similar elements were given specific names in order to distinguish them from the whole variety of expressions. In this article we will get acquainted with them, that is, we will give an overview of the basic expressions studied in algebra lessons at school.

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Monomials and polynomials

Let's start with expressions called monomials and polynomials. At the time of writing this article, the conversation about monomials and polynomials begins in 7th grade algebra lessons. The following definitions are given there.

Definition.

Monomials numbers, variables, their powers with a natural exponent, as well as any products composed of them are called.

Definition.

Polynomials is the sum of the monomials.

For example, the number 5, the variable x, the power z 7, the products 5 x and 7 x x 2 7 z 7 are all monomials. If we take the sum of monomials, for example, 5+x or z 7 +7+7·x·2·7·z 7, we get a polynomial.

Working with monomials and polynomials often involves doing things with them. Thus, on the set of monomials, the multiplication of monomials and the raising of a monomial to a power are defined, in the sense that as a result of their execution a monomial is obtained.

Addition, subtraction, multiplication, and exponentiation are defined on the set of polynomials. How these actions are determined and by what rules they are performed, we will talk in the article actions with polynomials.

If we talk about polynomials with a single variable, then when working with them, dividing a polynomial by a polynomial has significant practical significance, and often such polynomials have to be represented as a product; this action is called factoring the polynomial.

Rational (algebraic) fractions

In the 8th grade, the study of expressions containing division by an expression with variables begins. And the first such expressions are rational fractions, which some authors call algebraic fractions.

Definition.

Rational (algebraic) fraction is a fraction whose numerator and denominator are polynomials, in particular monomials and numbers.

Here are some examples of rational fractions: and . By the way, any ordinary fraction is a rational (algebraic) fraction.

On set algebraic fractions addition, subtraction, multiplication, division and exponentiation are introduced. How this is done is explained in the article Actions with algebraic fractions.

It is often necessary to perform transformations of algebraic fractions, the most common of which are reduction and reduction to a new denominator.

Rational Expressions

Definition.

Expressions with powers (power expressions) are expressions containing degrees in their notation.

Here are some examples of expressions with powers. They may not contain variables, for example, 2 3 , . Power expressions with variables also take place: and so on.

It wouldn't hurt to familiarize yourself with how it's done. converting expressions with powers.

Irrational expressions, expressions with roots

Definition.

Expressions containing logarithms are called logarithmic expressions.

Examples of logarithmic expressions are log 3 9+lne , log 2 (4 a b) , .

Very often, expressions contain both powers and logarithms, which is understandable, since by definition a logarithm is an exponent. As a result, expressions like this look natural: .

To continue the topic, refer to the material converting logarithmic expressions.

Fractions

In this section we will look at expressions of a special type - fractions.

The fraction expands the concept. Fractions also have a numerator and denominator located above and below the horizontal fraction line (to the left and right of the slanted fraction line), respectively. Only, unlike ordinary fractions, the numerator and denominator can contain not only natural numbers, but also any other numbers, as well as any expressions.

So, let's define a fraction.

Definition.

Fraction is an expression consisting of a numerator and a denominator separated by a fractional line, which represent some numerical or alphabetic expressions or numbers.

This definition allows you to give examples of fractions.

Let's start with examples of fractions whose numerators and denominators are numbers: 1/4, , (−15)/(−2) . The numerator and denominator of a fraction can contain expressions, both numerical and alphabetic. Here are examples of such fractions: (a+1)/3, (a+b+c)/(a 2 +b 2), .

But the expressions 2/5−3/7 are not fractions, although they contain fractions in their notations.

General expressions

In high school, especially in problems of increased difficulty and problems of group C in the Unified State Examination in mathematics, you will come across expressions complex type, containing in their notation simultaneously roots, powers, logarithms, and trigonometric functions, and so on. For example, or . They appear to fit several types of expressions listed above. But they are usually not classified as one of them. They are considered general expressions, and when describing they simply say an expression, without adding additional clarifications.

Concluding the article, I would like to say that if a given expression is cumbersome, and if you are not entirely sure what type it belongs to, then it is better to call it simply an expression than to call it an expression that it is not.

Bibliography.

• Mathematics: textbook for 5th grade. general education institutions / N. Ya. Vilenkin, V. I. Zhokhov, A. S. Chesnokov, S. I. Shvartsburd. - 21st ed., erased. - M.: Mnemosyne, 2007. - 280 pp.: ill. ISBN 5-346-00699-0.
• Mathematics. 6th grade: educational. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
• Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
• Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Algebra: 9th grade: educational. for general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2009. - 271 p. : ill. - ISBN 978-5-09-021134-5.
• Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Algebraic expression- this is any record of letters, numbers, arithmetic signs and brackets, composed with meaning. Essentially, an algebraic expression is a numerical expression in which, in addition to numbers, letters are also used. That's why algebraic expressions also called literal expressions.

Letters are mainly used in literal expressions Latin alphabet. What are these letters for? Instead we can substitute different numbers. That's why these letters are called variables. That is, they can change their meaning.

Examples of algebraic expressions.

$\begin(align) & x+5;\,\,\,\,\,(x+y)\centerdot (x-y);\,\,\,\,\,\frac(a-b)(2) ; \\ & \\ & \sqrt(((b)^(2))-4ac);\,\,\,\,\,\frac(2)(z)+\frac(1)(h); \,\,\,\,\,a((x)^(2))+bx+c; \\ \end(align)$

If, for example, in the expression x + 5 we substitute some number instead of the variable x, we will get a numerical expression. In this case, the value of this numerical expression will be the value of the algebraic expression x + 5 at given value variable. That is, for x = 10, x + 5 = 10 + 5 = 15. And for x = 2, x + 5 = 2 + 5 = 7.

There are values ​​of a variable at which the algebraic expression loses its meaning. This will happen, for example, if in the expression 1:x we substitute the value 0 instead of x.
Because you can't divide by zero.

The domain of definition of an algebraic expression.

The set of values ​​of a variable for which the expression does not lose meaning is called domain of definition this expression. We can also say that the domain of an expression is the set of all valid values ​​of a variable.

Let's look at examples:

1. y+5 – the domain of definition will be any values ​​of y.
2. 1:x – the expression will make sense for all values ​​of x except 0. Therefore, the domain of definition will be any values ​​of x except zero.
3. (x+y):(x-y) – domain of definition – any values ​​of x and y for which x ≠ y.

Types of algebraic expressions.

Rational algebraic expressions are integer and fractional algebraic expressions.

1. Whole algebraic expression – does not contain exponentiation with a fractional exponent, taking the root of a variable, or dividing by a variable. In integer algebraic expressions, all variable values ​​are valid. For example, ax + bx + c is an integer algebraic expression.
2. Fractional – contains division by a variable. $\frac(1)(a)+bx+c$ is a fractional algebraic expression. In fractional algebraic expressions, all variable values ​​that do not divide by zero are valid.

Irrational algebraic expressions contain taking the root of a variable or raising a variable to a fractional power.

$\sqrt(((a)^(2))+((b)^(2)));\,\,\,\,\,\,\,((a)^(\frac(2) (3)))+((b)^(\frac(1)(3)));$- irrational algebraic expressions. In irrational algebraic expressions, all values ​​of variables are valid for which the expression under the sign of an even root is not negative.

Numerical and algebraic expressions. Converting Expressions.

What is an expression in mathematics? Why do we need expression conversions?

The question, as they say, is interesting... The fact is that these concepts are the basis of all mathematics. All mathematics consists of expressions and their transformations. Not very clear? Let me explain.

Let's say you have an evil example in front of you. Very big and very complex. Let's say you're good at math and aren't afraid of anything! Can you give an answer right away?

You'll have to decide this example. Consistently, step by step, this example simplify. According to certain rules, of course. Those. do expression conversion. The more successfully you carry out these transformations, the stronger you are in mathematics. If you don't know how to do the right transformations, you won't be able to do them in math. Nothing...

To avoid such an uncomfortable future (or present...), it doesn’t hurt to understand this topic.)

First, let's find out what is an expression in mathematics. What's happened numeric expression and what is algebraic expression.

What is an expression in mathematics?

Expression in mathematics- this is a very broad concept. Almost everything we deal with in mathematics is a set of mathematical expressions. Any examples, formulas, fractions, equations, and so on - it all consists of mathematical expressions.

3+2 is a mathematical expression. c 2 - d 2- this is also a mathematical expression. And a healthy fraction, and even one number - that’s all mathematical expressions. For example, the equation is:

5x + 2 = 12

consists of two mathematical expressions connected by an equal sign. One expression is on the left, the other on the right.

IN general view term " mathematical expression"is used, most often, to avoid humming. They will ask you what an ordinary fraction is, for example? And how to answer?!

First answer: "This is... mmmmmm... such a thing... in which... Can I write a fraction better? Which one do you want?"

The second answer: “An ordinary fraction is (cheerfully and joyfully!) mathematical expression , which consists of a numerator and a denominator!"

The second option will be somehow more impressive, right?)

This is the purpose of the phrase " mathematical expression "very good. Both correct and solid. But for practical application need to be well versed in specific types of expressions in mathematics .

The specific type is another matter. This It's a completely different matter! Each type of mathematical expression has mine a set of rules and techniques that must be used when making a decision. For working with fractions - one set. For working with trigonometric expressions - the second one. For working with logarithms - the third. And so on. Somewhere these rules coincide, somewhere they differ sharply. But don't be afraid of these scary words. We will master logarithms, trigonometry and other mysterious things in the appropriate sections.

Here we will master (or - repeat, depending on who...) two main types of mathematical expressions. Numerical expressions and algebraic expressions.

Numeric expressions.

What's happened numeric expression? This is a very simple concept. The name itself hints that this is an expression with numbers. That is how it is. A mathematical expression made up of numbers, brackets and arithmetic symbols is called a numerical expression.

7-3 is a numerical expression.

(8+3.2) 5.4 is also a numerical expression.

And this monster:

also a numerical expression, yes...

An ordinary number, a fraction, any example of calculation without X's and other letters - all these are numerical expressions.

Main sign numerical expressions - in it no letters. None. Only numbers and mathematical symbols (if necessary). It's simple, right?

And what can you do with numerical expressions? Numeric expressions can usually be counted. To do this, it happens that you have to open the brackets, change signs, abbreviate, swap terms - i.e. do expression conversions. But more on that below.

Here we will deal with such a funny case when with a numerical expression you don't need to do anything. Well, nothing at all! This pleasant operation - To do nothing)- is executed when the expression doesn't make sense.

When does a numerical expression make no sense?

It’s clear that if we see some kind of abracadabra in front of us, like

then we won’t do anything. Because it’s not clear what to do about it. Some kind of nonsense. Maybe count the number of pluses...

But there are outwardly quite decent expressions. For example this:

(2+3) : (16 - 2 8)

However, this expression also doesn't make sense! For the simple reason that in the second brackets - if you count - you get zero. But you can’t divide by zero! This is a forbidden operation in mathematics. Therefore, there is no need to do anything with this expression either. For any task with such an expression, the answer will always be the same: "The expression has no meaning!"

To give such an answer, of course, I had to calculate what would be in brackets. And sometimes there’s a lot of stuff in parentheses... Well, there’s nothing you can do about it.

There are not so many forbidden operations in mathematics. There is only one in this topic. Division by zero. Additional restrictions arising in roots and logarithms are discussed in the corresponding topics.

So, an idea of ​​what it is numeric expression- got. Concept the numeric expression doesn't make sense- realized. Let's move on.

Algebraic expressions.

If letters appear in a numerical expression, this expression becomes... The expression becomes... Yes! It becomes algebraic expression. For example:

5a 2; 3x-2y; 3(z-2); 3.4m/n; x 2 +4x-4; (a+b) 2; ...

Such expressions are also called literal expressions. Or expressions with variables. It's practically the same thing. Expression 5a +c, for example, both literal and algebraic, and an expression with variables.

Concept algebraic expression - broader than numeric. It includes and all numerical expressions. Those. a numerical expression is also an algebraic expression, only without letters. Every herring is a fish, but not every fish is a herring...)

Why alphabetic- It's clear. Well, since there are letters... Phrase expression with variables It’s also not very puzzling. If you understand that numbers are hidden under the letters. All sorts of numbers can be hidden under letters... And 5, and -18, and whatever you want. That is, a letter can be replace on different numbers. That's why the letters are called variables.

In expression y+5, For example, at- variable value. Or they just say " variable", without the word "magnitude". Unlike five, which is a constant value. Or simply - constant.

Term algebraic expression means that to work with this expression you need to use laws and rules algebra. If arithmetic works with specific numbers, then algebra- with all the numbers at once. A simple example for clarification.

In arithmetic we can write that

But if we write such an equality through algebraic expressions:

a + b = b + a

we'll decide right away All questions. For all numbers stroke. For everything infinite. Because under the letters A And b implied All numbers. And not only numbers, but even other mathematical expressions. This is how algebra works.

When does an algebraic expression not make sense?

Everything about the numerical expression is clear. You can't divide by zero there. And with letters, is it possible to find out what we are dividing by?!

Let's take for example this expression with variables:

2: (A - 5)

Does it make sense? Who knows? A- any number...

Any, any... But there is one meaning A, for which this expression exactly doesn't make sense! And what is this number? Yes! This is 5! If the variable A replace (they say “substitute”) with the number 5, in brackets you get zero. Which cannot be divided. So it turns out that our expression doesn't make sense, If a = 5. But for other values A does it make sense? Can you substitute other numbers?

Certainly. In such cases they simply say that the expression

2: (A - 5)

makes sense for any values A, except a = 5 .

The whole set of numbers that Can substituting into a given expression is called range of acceptable values this expression.

As you can see, there is nothing tricky. Let's look at the expression with variables and figure out: at what value of the variable is the forbidden operation (division by zero) obtained?

And then be sure to look at the task question. What are they asking?

doesn't make sense, our forbidden meaning will be the answer.

If you ask at what value of a variable the expression has the meaning(feel the difference!), the answer will be all other numbers except for what is forbidden.

Why do we need the meaning of the expression? He is there, he is not... What's the difference?! The point is that this concept becomes very important in high school. Extremely important! This is the basis for such solid concepts as the domain of acceptable values ​​or the domain of a function. Without this, you will not be able to solve serious equations or inequalities at all. Like this.

Converting Expressions. Identity transformations.

We were introduced to numerical and algebraic expressions. We understood what the phrase “the expression has no meaning” means. Now we need to figure out what it is expression conversion. The answer is simple, to the point of disgrace.) This is any action with an expression. That's all. You have been doing these transformations since first grade.

Let's take the cool numerical expression 3+5. How can it be converted? Yes, very simple! Calculate:

This calculation will be the transformation of the expression. You can write the same expression differently:

Here we didn’t count anything at all. Just wrote down the expression in a different form. This will also be a transformation of the expression. You can write it like this:

And this too is a transformation of an expression. You can make as many such transformations as you want.

Any action on expression any writing it in another form is called transforming the expression. And that's all. Everything is very simple. But there is one thing here very important rule. So important that it can safely be called main rule all mathematics. Breaking this rule inevitably leads to errors. Are we getting into it?)

Let's say we transformed our expression haphazardly, like this:

Conversion? Certainly. We wrote the expression in a different form, what’s wrong here?

It's not like that.) The point is that transformations "at random" are not interested in mathematics at all.) All mathematics is built on transformations in which appearance, but the essence of the expression does not change. Three plus five can be written in any form, but it must be eight.

Transformations, expressions that do not change the essence are called identical.

Exactly identity transformations and allow us, step by step, to transform a complex example into a simple expression, while maintaining the essence of the example. If we make a mistake in the chain of transformations, we make a NOT identical transformation, then we will decide another example. With other answers that are not related to the correct ones.)

This is the main rule for solving any tasks: maintaining the identity of transformations.

I gave an example with the numerical expression 3+5 for clarity. In algebraic expressions, identity transformations are given by formulas and rules. Let's say in algebra there is a formula:

a(b+c) = ab + ac

This means that in any example we can instead of the expression a(b+c) feel free to write an expression ab + ac. And vice versa. This identical transformation. Mathematics gives us a choice between these two expressions. And which one to write - from concrete example depends.

Another example. One of the most important and necessary transformations is the basic property of a fraction. You can look at the link for more details, but here I’ll just remind you of the rule: If the numerator and denominator of a fraction are multiplied (divided) by the same number, or an expression that is not equal to zero, the fraction will not change. Here is an example of identity transformations using this property:

As you probably guessed, this chain can be continued indefinitely...) A very important property. It is this that allows you to turn all sorts of example monsters into white and fluffy.)

There are many formulas defining identical transformations. But the most important ones are quite a reasonable number. One of the basic transformations is factorization. It is used in all mathematics - from elementary to advanced. Let's start with him. In the next lesson.)

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.