Let's consider a small problem that is often found in various magazines and magic tricks.

The magician asks you to guess a certain number. Then he asks to multiply it by three, and add six to the result. Then he asks to divide the received amount by three and subtract the resulting number from the result. He then tells you the correct answer.

How does this happen, is it really magic?

No, it's actually easier. Let us think of the number 5. Now we will perform all the actions that the magician offered us.

• 1. 5*3=15.
• 2. 15+6=21.
• 3. 21:3=7.
• 4. 7-5=2.

We got two in response. We could write the same solution as a numerical expression (5 * 3 + 6): 3 - 5. And its value would be the number 2.

Now, let's say we conceived the number 3. The result would be a numerical expression (3 * 3 + 6): 3 - 3. And its value would be the number 2.

Two again. The thought arises that there is no trick here, and in any case the number 2 will be obtained. Let's try to check this. Let's denote the number we have conceived with the letter x, and write down all the actions that the magician asked to do in the required order.

• We get (x * 3 + 6): 3 -x.

• (x * 3 + 6): 3 - x \u003d x + 2-x \u003d 2.

It turns out that the number conceived by us does not play any role at all, it will be reduced in any case.

In the analysis of the problem, we received the expression (x * 3 + 6): 3 - x, which is written using a letter denoting any number, the numbers 3 and 6, brackets and action signs. Such an expression is called an algebraic expression or an expression with a variable.

Defining an expression with a variable

• An algebraic expression or an expression with a variable is called any meaningful notation consisting of letters denoting any number, numbers and action signs.

For example, the following entries would be algebraic expressions:

• 2*(x+y),
• 34*a-13*a*x,
• (123-65 * a): 3 +4.

If instead of each letter that is included in the algebraic expression, substitute a certain numerical value, and then perform all the actions, then the result will be a certain number. This number is called the value of an algebraic expression.

For example, the value of the algebraic expression 5*a+2*x-7 with a=2 and x=3 will be the number 9, since 5*2+2*3 -7 = 9.

In the problem that we considered at the beginning, the value of the algebraic expression (x * 3 + 6): 3 - x will be the number 2, for any value of the variable x.

Solving problems and some expressions does not always lead to clean numerical answers. Even in the case of trivial calculations, one can arrive at a certain construction, called an expression with a variable.

For example, consider two practical problems. In the first case, we have a plant that produces 5 tons of milk every day. It is necessary to find how much milk is produced by the plant in p days.

In the second case, there is a rectangle whose width is 5 cm and length p cm. Find the area of ​​the figure.

Of course, if a plant produces five tons per day, then in r days, according to the simplest mathematical logic, it will produce 5r tons of milk. On the other hand, the area of ​​a rectangle is equal to the product of its sides - that is, in this case, it is 5p. In other words, in two trivial problems with different conditions, the answer is one whole expression - 5p. Such monomials are called an expression with a variable, since in addition to the numeric part they contain some letter, called an unknown, or a variable. Such an element is denoted by lowercase letters of the Latin alphabet, most often, x or y, although this is not important.

A feature of a variable is that it can take on any value in practice. Substituting different numbers, we will get the final solution for our tasks, for example, for the first one:

p = 2 days, the plant produces 5p = 10 tons of milk;

p = 4 days, the plant produces 5p = 20 tons of milk;

Or for the second one:

p \u003d 10 cm, the area of ​​\u200b\u200bthe figure is 5p \u003d 50 cm2

p \u003d 20 cm, the area of ​​\u200b\u200bthe figure is 5p \u003d 100 cm2

It is important to understand that p is not a set of some individual values, but the whole set that will mathematically correspond to the condition of the problem. The main role of a variable is to replace the missing element in a condition. Any mathematical problem must include some constructions and display the relationship between these constructions in the condition. If the value of any object is not enough, then a variable is introduced instead. At the same time, it is an abstract replacement of the very element of the condition (the amount of something represented by a number or expression), and not by functional connections.

If we consider an expression of the form 5p as a neutral and independent object, then the value of p in it can take on any values, in fact, p here is equal to the set of all real numbers.

But in our problems, certain mathematical restrictions are imposed on the answer in the form of 5p, which follow from the conditions. For example, days and days cannot be negative, so p in both problems is always equal to or greater than zero. In addition, days cannot be fractional - for the first task, only those p-values ​​that are positive integers are valid.

In the first problem: p is equal to the finite set of all positive integers;

In the second problem: p is equal to the finite set of all positive numbers.

Expressions can include two variables at once, for example:

In this case, the binomial is represented by two monomials, each of which has a variable in its composition, and these variables are different, that is, independent of each other. The value of this expression can be fully calculated only if the value of both variables is present. For example, if x = 2 and y = 4, then:

2x + 3y \u003d 4 + 12 \u003d 16 (for x \u003d 2, y \u003d 4)

It is worth noting that in this expression there are no mathematical or logical restrictions on the values ​​of the variable - both x and y belong to the entire set of real numbers.

In general terms, the set of all numbers, when substituting for a variable, the expression retains meaning and validity, is called the domain of definition (or value) of the variable.

In abstract examples that are not related to real problems, the scope of a variable is most often either equal to the entire set of real numbers or limited to some constructions, for example, a fraction. As you know, when the divisor is zero, the whole fraction loses its meaning. Therefore, a variable in an expression of the form:

cannot be equal to five, because then:

7x / (x - 5) \u003d 7x / 0 (for x \u003d 5)

And the fraction will lose its meaning. Therefore, for this expression, the variable x has a domain of definition - the set of all numbers except 5.

In our video tutorial, a special case of using variables is also noted, when they denote a number of the same order. For example, the numbers 54, 30, 78 can be specified through the variable a, or through the construction ab (with a horizontal bar on top, to distinguish from the product), where b specifies units (4, 0, 8, respectively), and tens (respectively, 5, 3, 7).

Entries 2 A + 8, 3A + 5b, A 4 – bc are called expressions with variables. Substituting numbers instead of letters, we get numerical expressions. The general concept of an expression with variables is defined in exactly the same way as the concept of a numeric expression, only, in addition to numbers, expressions with variables can also contain letters.

For expressions with a variable, simplifications are also applied: do not put brackets containing only a number or a letter, do not put a multiplication sign between letters, between numbers and letters, etc.

There are expressions with one, two, three, etc. variables. designate A(X), IN(x, y) etc.

An expression with a variable cannot be called either a statement or a predicate. For example, about expression 2 A+ 5 it is impossible to say whether it is true or false, therefore, it is not a proposition. If instead of a variable A substitute the numbers, then we get various numerical expressions, which are also not statements, therefore, this expression is also not a predicate.

Each expression with a variable corresponds to a set of numbers, substituting which results in a numerical expression that makes sense. This set is called the domain of the expression.

Example. 8: (4 – X) - domain R\(4), because at X= 4 expression 8: (4 - 4) does not make sense.

If the expression contains multiple variables, for example, X And at, then the domain of this expression is the set of pairs of numbers ( A; b) such that when replacing X on A And at on b results in a numeric expression that has a value.

Example. , the domain of definition is the set of pairs ( A; b) │A≥ b.

Definition. Two expressions with a variable are called identically equal if for any values. Variables from the scope of expressions, their respective values ​​are equal.

That. two expressions A(X), IN(X) are identically equal on the set X, If

1) the sets of admissible values ​​of the variable in these expressions are the same;

2) for any X 0 their set of allowable values, the values ​​of expressions at X 0 match, i.e. A(X 0) = IN(X 0) is the correct numerical equality.

Example. (2 X+ 5) 2 and 4 X 2 + 20X+ 25 – identically equal expressions.

designate A(X) º IN(X). Note that if two expressions are identically equal on some set E, then they are identically equal on any subset E 1 M E. It should also be noted that the statement about the identical equality of two expressions with a variable is a statement.

If two expressions that are identically equal on a certain set are joined by an equal sign, then we get a sentence, which is called an identity on this set.

True numerical equalities are also considered identities. Identities are the laws of addition and multiplication of real numbers, the rules for subtracting a number from a sum and a sum from a number, the rules for dividing a sum by a number, etc. Identities are also rules for operations with zero and one.

Replacing an expression with another one that is identical to it on some set is called the identical transformation of the given expression.

Example. 7 X + 2 + 3X = 10 X+ 2 - identical transformation, not identical transformation on R.

§ 5. Classification of expressions with a variable

1) An expression composed of variables and numbers using only the operations of addition, subtraction, multiplication, exponentiation is called an integer expression or a polynomial.

Example. (3X 2 + 5) ∙ (2X – 3at)

2) Rational is an expression built from variables and numbers using the operations of addition, subtraction, multiplication, division, exponentiation. A rational expression can be represented as a ratio of two integer expressions, i.e. polynomials. Note that integer expressions are a special case of rational ones.

Example. .

3) Irrational is an expression built from variables and numbers using the operations of addition, subtraction, multiplication, division, exponentiation, as well as the operation of extracting the root P-th degree.

In algebra lessons at school, we come across expressions of various kinds. As you learn new material, expressions become more diverse and more complex. For example, we got acquainted with degrees - degrees appeared in the composition of expressions, we studied fractions - fractional expressions appeared, etc.

For the convenience of describing the material, expressions consisting of similar elements were given certain 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.

Page navigation.

Monomials and polynomials

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

Definition.

monomials called numbers, variables, their degrees with a natural indicator, as well as any products made up of them.

Definition.

Polynomials is the sum of monomials.

For example, the number 5 , the variable x , the degree z 7 , the products 5 x and 7 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 , then we get a polynomial.

Working with monomials and polynomials often means doing things with them. So, 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.

On the set of polynomials, addition, subtraction, multiplication, exponentiation are defined. How these actions are defined, 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, division of a polynomial by a polynomial is of considerable practical importance, and often such polynomials have to be represented as a product, this action is called factorization of a polynomial.

Rational (algebraic) fractions

In grade 8, 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 it 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.

Addition, subtraction, multiplication, division and exponentiation are introduced on the set of algebraic fractions. How this is done is explained in the article Operations with Algebraic Fractions.

Often you have to perform transformations of algebraic fractions, the most common of which are reduction and reduction to a new denominator.

Rational Expressions

Definition.

Power expressions (power expressions) are expressions containing degrees in their notation.

Here are some examples of expressions with powers. They may not contain variables, such as 2 3 , . There are also power expressions with variables: and so on.

It doesn't hurt to get familiar with how transformation of 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 in expressions both degrees and logarithms occur at the same time, which is understandable, since, by definition, a logarithm is an exponent. As a result, expressions of this kind look natural: .

Continuing the topic, refer to the material transformation of logarithmic expressions.

Fractions

In this paragraph, we will consider expressions of a special kind - fractions.

The fraction expands the concept. Fractions also have a numerator and denominator located above and below the horizontal fractional bar (left and right of the slash), 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 bar, which represent some numeric or alphabetic expression or number.

This definition allows us 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 records.

General expressions

In high school, especially in tasks of increased difficulty and tasks of group C in the USE in mathematics, expressions of a complex form will come across, containing in their record both roots, powers, logarithms, and trigonometric functions, etc. For example, or . They seem 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 just say an expression, without adding additional clarifications.

Concluding the article, I would like to say that if this expression is cumbersome, and if you are not quite sure what kind it belongs to, then it is better to call it just an expression than to call it such an expression as it is not.

Bibliography.

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

Find the value of the expression x+5 if x=0, x=3, x=16, x=35

We reason like this:

if x=0, then the value of the sum is 5, because 0+5=5

if x=3, then the value of the sum is 8, because 3+5=8

if x=16, then the value of the sum is 21, since 16+5=21

if x=35, then the value of the sum is 40, since 35+5=40

What other values ​​can x take?

X can be 43 or 68. In general, you can say that x can take on any value.

What would you name a letter that can take on any value?

You can call it in different ways: changeable, changeable.

Correct answer: in mathematics it is called a variable.

Please note: in mathematics, a variable allows you to write several expressions in one.

Let's consider expressions. What can be said about them?

The correct answer is: the minuends are the same, but the subtrahends change. So, it can be written like this:

Consider expressions with a variable.

What common? What is the difference?

Correct answer: in all expressions there is one action, in all expressions there is a number 2. Differences: different actions, different letters denote a variable.

What values ​​can the variable take in these expressions?

In the expression 2+x, x can be any number.

In the expression 2*y, y can be any number.

In the 2-z expression, z can only take on a few values: z=2, z=1, z=0.

Today in the lesson we repeated the differences between a simple and a compound problem, remembered how to add and subtract two-digit numbers in a column.

Let's find the value of these expressions if x=5, y=3, z=2.

We argue as follows: we substitute these numbers into expressions.

If x=5 then 2+x=2+5=7

If y=3 then 2*y=2*3=6

If z=2 then 2-z=2-2=0

Read and compare tasks.

1. Tanya has 3 roses and 6 peonies. How many flowers does Tanya have?

2. Tanya has 3 roses and 4 peonies. How many flowers does Tanya have?

3. Tanya has 3 roses and 2 peonies. How many flowers does Tanya have?

Let's pay attention to the fact that the number of peony flowers changes in the problem. Let's replace all three tasks with one task with a variable. Then the problem will sound like this: Tanya has 3 roses and k peonies. How many flowers does Tanya have?

To find out how many flowers Tanya has, you need to add k to 3.

Substitute the values ​​into the literal expression.

if k=6 3+6=9 (color)

if k=4 3+4=7 (color)

if k=2 3+2=5 (color)

It is important to note that sometimes there are two variables in an expression.

Then the expressions might look like this:

Determine which variable is larger and by how much.

Correct answer:

in the first equality, we compare the variables b and a, a is the result of addition, so a>b by 18;

in the second equality, we compare the variables n and m, n is decreasing, which means n>m by 4;

in the third equality, we compare the variables c and d, c is the term, d is the value of the sum, which means d>c by 7;

in the fourth equality k-t =5 we compare the minuend and the subtrahend, the minuend is greater, therefore k>t by 5.

Today in the lesson we learned to compose expressions with a variable, found the values ​​of expressions for a given value of the variable.

Bibliography

1. M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
2. M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
3. M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
4. Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
5. "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
6. S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
7. V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.

1. Nsportal.ru ().
2. Prosv.ru ().
3. Do.gendocs.ru ().

Homework

1. Find the value of the expression 36 - a, if a \u003d 15, a \u003d 16, a \u003d 20, a \u003d 35.

2. Find the value of the expression 12 + x, if x = 10, x = 34, x = 48, x = 59

3. Compare the expressions with the variable and put a comparison sign. 36 + k ... 37 + k

4. Replace these expressions with one common with a variable.