Determine which concepts are not identical. The meaning of the word identity

Let's start talking about identities, give a definition of the concept, introduce notations, and consider examples of identities.

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What is identity

Let's start with the definition of the concept of identity.

Definition 1

An identity is an equality that is true for any values ​​of the variables. In fact, any numerical equality is an identity.

As we analyze the topic, we can clarify and supplement this definition. For example, if we recall the concepts of permissible values ​​of variables and ODZ, then the definition of identity can be given in the following way.

Definition 2

Identity is a true numerical equality, as well as an equality that will be true for all acceptable values variables that make up it.

Any values ​​of variables when determining an identity are discussed in manuals and textbooks on mathematics for the 7th grade, since school program for seventh graders involves performing actions exclusively with whole expressions (mono- and polynomials). They make sense for any values ​​of the variables that comprise them.

The 8th grade program is expanded by considering expressions that make sense only for the values ​​of variables from the DL. In this regard, the definition of identity changes. In fact, identity becomes a special case of equality, since not every equality is an identity.

Identity sign

Notation of equality assumes the presence of an equal sign “=”, from which some numbers or expressions are located to the right and left. The identity sign looks like three parallel lines"≡". It is also called the identical equals sign.

Typically, writing an identity is no different from writing an ordinary equality. The sign of identity can be used to emphasize that this is not simple equality, but identity.

Examples of identities

Let's look at some examples.

Example 1

Numerical Equalities 2 ≡ 2 and - 3 ≡ - 3 are examples of identities. According to the definition given above, any true numerical equality is by definition an identity, and the given equalities are true. They can also be written as follows 2 ≡ 2 and - 3 ≡ - 3 .

Example 2

Identities can contain not only numbers, but also variables.

Example 3

Let's take equality 3 (x + 1) = 3 x + 3. This equality is true for any value of the variable x. This fact is confirmed by the distributive property of multiplication relative to addition. This means that the given equality is an identity.

Example 4

Let's take the identity y · (x − 1) ≡ (x − 1) · x: x · y 2: y . Let's consider the range of permissible values ​​of the variables x and y. These are any numbers except zero.

Example 5

Take the equalities x + 1 = x − 1, a + 2 b = b + 2 a And | x | = x. There are a number of variable values ​​for which these equalities are not true. For example, when x = 2 equality x + 1 = x − 1 turns into false equality 2 + 1 = 2 − 1 . And in general, equality x + 1 = x − 1 is not achieved for any value of the variable x.

In the second case the equality a + 2 b = b + 2 a false in any case where the variables a and b have different meanings. Let's take a = 0 And b = 1 and we get an incorrect equality 0 + 2 1 = 1 + 2 0.

Equality in which | x |- the modulus of the variable x is also not an identity, since it is not true for negative values ​​of x.

This means that the given equalities are not identities.

Example 6

In mathematics we constantly deal with identities. By recording actions performed with numbers, we work with identities. Identities are records of properties of powers, properties of roots, and others.

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Explanatory dictionary of the Russian language. S.I.Ozhegov, N.Yu.Shvedova.

identity

And also IDENTITY. -a, cf.

Complete similarity, coincidence. G. views.

(identity). In mathematics: an equality that is valid for any numerical values ​​of the quantities included in it. || adj. identical, -aya, -oe and identical, -aya, -oe (to 1 meaning). Identical algebraic expressions. ALSO [not to be confused with the combination of the pronoun “that” and the particle “same”].

1. adv. In the same way, just like whoever. You're tired, I'm...

   union. Same as also. You're leaving, and brother? - T.

particle. Expresses a distrustful or negative, ironic attitude (simple). *T. I've found a smart guy! He is a poet. - Poet t. (to me)!

New explanatory dictionary of the Russian language, T. F. Efremova.

identity

1. Absolute coincidence with someone or something. both in its essence and in external signs and manifestations.

   Exact match of smth. something

Wed

An equality that is valid for all numerical values ​​of the letters included in it (in mathematics).

identity

the relationship between objects (objects of reality, perception, thought) considered as “one and the same”; "limiting" case of equality relation. In mathematics, an identity is an equation that is satisfied identically, i.e. is valid for any admissible values ​​of the variables included in it.

Identity

basic concept of logic, philosophy and mathematics; used in languages scientific theories for the formulation of defining relations, laws and theorems. In mathematics, T. ≈ is an equation that is satisfied identically, that is, valid for any permissible values ​​of the variables included in it. From a logical point of view, T. ≈ is a predicate represented by the formula x = y (read: “x is identical to y”, “x is the same as y”), to which corresponds a logical function, true when the variables x and y mean different occurrences of the “same” object, and false otherwise. From a philosophical (epistemological) point of view, T. is a relationship based on ideas or judgments about what the “same” object of reality, perception, and thought is. The logical and philosophical aspects of theory are complementary: the first provides a formal model of the concept of theory, and the second provides the basis for applying this model. The first aspect includes the concept of the “same” object, but the meaning of the formal model does not depend on the content of this concept: the identification procedures and the dependence of the identification results on the conditions or methods of identification, on the abstractions explicitly or implicitly accepted in this case are ignored. In the second (philosophical) aspect of consideration, the grounds for the use of logical models of T. are associated with how objects are identified, by what criteria, and already depend on the point of view, on the conditions and means of identification. The distinction between the logical and philosophical aspects of theory goes back to the well-known position that a judgment about the identity of objects and theory as a concept is not the same thing (see Plato, Soch., vol. 2, Moscow, 1970, p. 36) . It is important, however, to emphasize the independence and consistency of these aspects: the concept of T. is exhausted by the meaning of the logical function corresponding to it; it is not derived from the actual identity of objects, “is not extracted” from it, but is an abstraction, replenished in “suitable” conditions of experience or, in theory, through assumptions (hypotheses) about actually permissible identifications; at the same time, when substitution is fulfilled (see below axiom 4) in the corresponding interval of abstraction of identification, “within” this interval, the actual T. of objects exactly coincides with T. in the logical sense. The importance of the concept of T. has determined the need for special theories T. The most common way of constructing these theories is axiomatic. As axioms, you can specify, for example, the following (not necessarily all):

x = y É y = x,

x = y & y = z É x = z,

A (x) É (x = y É A (y)),

where A (x) ≈ an arbitrary predicate containing x freely and free for y, and A (x) and A (y) differ only in the occurrences (at least one) of the variables x and y.

Axiom 1 postulates the property of reflexivity of T. In traditional logic, it was considered the only logical law of T., to which axioms 2 and 3 were usually added as “non-logical postulates” (in arithmetic, algebra, geometry). Axiom 1 can be considered epistemologically justified, since it is kind of logical expression individuation, on which, in turn, the “givenness” of objects in experience, the possibility of their recognition, is based: in order to talk about an object “as given,” it is necessary to somehow highlight it, distinguish it from other objects and not confuse it with them in the future. In this sense, T., based on axiom 1, is a special relation of “self-identity” that connects each object only with itself and with no other object.

Axiom 2 postulates the property of symmetry T. It asserts the independence of the result of identification from the order in pairs of identified objects. This axiom also has a well-known justification in experience. For example, the order of weights and goods on a scale is different from left to right for the buyer and seller facing each other, but the result - equilibrium in this case - is the same for both.

Axioms 1 and 2 together serve as an abstract expression of theory as indistinguishability, a theory in which the idea of ​​the “same” object is based on the facts of the non-observability of differences and significantly depends on the criteria of distinguishability, on the means (instruments) that distinguish one object from another , ultimately ≈ from the abstraction of indiscernibility. Since the dependence on the “distinction threshold” is fundamentally irremovable in practice, the idea of ​​a T that satisfies axioms 1 and 2 is the only natural result that can be obtained in an experiment.

Axiom 3 postulates the transitivity of T. It states that the superposition of T. is also T. and is the first non-trivial statement about the identity of objects. Transitivity of T. is either an “idealization of experience” under conditions of “decreasing accuracy”, or an abstraction that replenishes experience and “creates” a new, different from indiscernibility, meaning of T.: indistinguishability guarantees only T. in the interval of abstraction of indistinguishability, and this latter does not is associated with the fulfillment of Axiom 3. Axioms 1, 2, and 3 together serve as an abstract expression of the theory of T. as equivalence.

Axiom 4 postulates a necessary condition for T. objects the coincidence of their characteristics. From a logical point of view, this axiom is obvious: all its attributes belong to the “same” object. But since the idea of ​​the “same” thing is inevitably based on certain kinds of assumptions or abstractions, this axiom is not trivial. It cannot be verified “in general” - according to all conceivable signs, but only in certain fixed intervals of abstractions of identification or indistinguishability. This is exactly how it is used in practice: objects are compared and identified not according to all conceivable characteristics, but only according to some ≈ basic (initial) characteristics of the theory in which they want to have a concept of the “same” object based on these characteristics and on axiom 4. In these cases, the scheme of axioms 4 is replaced by a finite list of its alloforms ≈ “meaningful” axioms T congruent to it. For example, in axiomatic set theory Zermelo ≈ Frenkel ≈ axioms:

4.1 z Î x É (x = y É z Î y),

4.2 x Î z É (x = y É y Î z),

defining, provided that the universe contains only sets, the interval of abstraction of identification of sets by “membership in them” and by their “own membership”, with the obligatory addition of axioms 1≈3, defining T. as equivalence.

The axioms 1≈4 listed above belong to the so-called laws of T. From them, using the rules of logic, many other laws that are unknown in pre-mathematical logic can be derived. The distinction between the logical and epistemological (philosophical) aspects of theory does not matter as long as we are talking about general abstract formulations of the laws of theory. The matter, however, changes significantly when these laws are used to describe realities. By defining the concept of “one and the same” object, the axiomatics of theory necessarily influence the formation of the universe “within” the corresponding axiomatic theory.

Lit.: Tarski A., Introduction to logic and methodology of deductive sciences, trans. from English, M., 1948; Novoselov M., Identity, in the book: Philosophical Encyclopedia, vol. 5, M., 1970; by him, On some concepts of the theory of relations, in the book: Cybernetics and modern scientific knowledge, M., 1976; Shreider Yu. A., Equality, similarity, order, M., 1971; Kleene S.K., Mathematical Logic, trans. from English, M., 1973; Frege G., Schriften zur Logik, B., 1973.

M. M. Novoselov.

Wikipedia

Identity (mathematics)

Identity(in mathematics) - an equality that holds for the entire set of values ​​of the variables included in it, for example:

etc. Sometimes an equality that does not contain any variables is also called an identity; eg 25 = 625.

Identical equality, when they want to emphasize it especially, is denoted by the symbol “ ≡ ”.

Identity

Identity, identity- ambiguous terms.

• Identity is an equality that holds for the entire set of values ​​of the variables included in it.
• Identity is the complete coincidence of the properties of objects.
• Identity in physics is a characteristic of objects in which replacing one of the objects with another does not change the state of the system while maintaining the given conditions.
• The law of identity is one of the laws of logic.
• The principle of identity is a principle of quantum mechanics, according to which the states of a system of particles, obtained from each other by rearranging identical particles in places, cannot be distinguished in any experiment, and such states must be considered as one physical state.
• “Identity and Reality” - book by E. Meyerson.

Identity (philosophy)

Identity- a philosophical category expressing equality, the sameness of an object, a phenomenon with itself, or the equality of several objects. Objects A and B are said to be identical, the same, if and only if all properties. This means that identity is inextricably linked with difference and is relative. Any identity of things is temporary, transitory, but their development and change is absolute. In the exact sciences, however, abstract, that is, identity abstracted from the development of things, in accordance with Leibniz’s law, is used because in the process of cognition, idealization and simplification of reality are possible and necessary under certain conditions. The logical law of identity is formulated with similar restrictions.

Identity should be distinguished from similarity, similarity and unity.

We call similar objects that have one or more common properties; the more items have general properties, the closer their similarity comes to identity. Two objects are considered identical if their qualities are completely similar.

However, it should be remembered that in the objective world there cannot be identity, since two objects, no matter how similar they are in quality, still differ in number and the space they occupy; only where material nature is elevated to spirituality does the possibility of identity arise.

A necessary condition for identity is unity: where there is no unity, there cannot be identity. The material world, divisible to infinity, does not have unity; unity comes with life, especially with spiritual life. We speak of the identity of an organism in the sense that its single life persists despite the constant change of particles that form the organism; where there is life, there is unity, but in the real meaning of the word there is still no identity, since life waxes and wanes, remaining unchanged only in the idea.

The same can be said about personalities- the highest manifestation of life and consciousness; and in personality we only assume identity, but in reality there is none, since the very content of personality is constantly changing. True identity is possible only in thinking; a correctly formed concept has eternal value regardless of the conditions of time and space in which it is thought.

Leibniz, with his principium indiscernibilium, established the idea that two things cannot exist that are completely similar in qualitative and quantitative respects, since such similarity would be nothing more than identity.

The philosophy of identity is a central idea in the works of Friedrich Schelling.

Examples of the use of the word identity in literature.

This is precisely the great psychological merit of both ancient and medieval nominalism, that it thoroughly dissolved the primitive magical or mystical identity words with an object - too solid even for the type whose basis is not in holding tightly to things, but in abstracting the idea and placing it above things.

This identity subjectivity and objectivity and constitutes precisely the universality now achieved by self-consciousness, rising above both mentioned sides, or particularities, and dissolving them in itself.

At this stage, self-conscious subjects correlated with each other have risen, therefore, through the removal of their unequal particularity of individuality to the consciousness of their real universality - the inherent freedom of all of them - and thereby to the contemplation of a certain identities them with each other.

A century and a half later, Inta, the great-great-great-granddaughter of the woman to whom he gave way to spaceship Sarp, amazed by her inexplicable identity with Vella.

But when it turned out that before his death, the good writer Kamanin read the manuscript of KRASNOGOROV and, at the same time, the same one whose candidacy was discussed by the fierce physicist Sherstnev a second before his, Sherstnev, SIMILAR death - here, you know, something not simple smelled to me anymore Coincidentally, there was a smell IDENTITY!

Klossowski's merit is that he showed that these three forms are now linked forever, but not through dialectical transformation and identity opposites, but thanks to their dispersion on the surface of things.

In these works, Klossowski develops a theory of sign, meaning and nonsense, and also gives a deeply original interpretation of Nietzsche’s idea of ​​eternal recurrence, understood as an eccentric ability to affirm divergences and disjunctions, leaving no room for either identity I neither identity peace or identity God.

As in any other type of identification of a person based on appearance, in photographic examination the identified object in all cases is a specific individual, identity which is installed.

Now the teacher has emerged from the student, and first of all, as a teacher, he coped with the great task of the first period of his master's degree, winning victory in the struggle for authority and complete identity person and position.

But in the early classics it is identity the thinker and the thinkable were interpreted only intuitively and only descriptively.

For Schelling identity Nature and Spirit is a natural philosophical principle that precedes empirical knowledge and determines the understanding of the results of the latter.

Based on this identities mineral characteristics and concluded that this Scottish formation is contemporary with the lowermost Wallis formations, because the amount of paleontological data available is too small to support or refute such a position.

Now it is no longer the origin that gives place to historicity, but the very fabric of historicity reveals the need for a origin that would be both internal and external, like some hypothetical apex of a cone, where all differences, all dispersion, all discontinuities are compressed into a single point identities, into that disembodied image of the Same, capable, however, of splitting and turning into the Other.

It is known that there are often cases when an object to be identified from memory does not have a sufficient number of noticeable features that would allow it to be identified identity.

It is clear, therefore, that there should be no vehements, or uprisings, in Moscow against people who wanted to flee from the Tatars, in Rostov against the Tatars, in Kostroma, Nizhny, Torzhok against the boyars, veches convened by all the bells, one by one identity names, to be confused with the veches of Novgorod and other old cities: Smolensk, Kyiv, Polotsk, Rostov, where the inhabitants, according to the chronicler, as if in a Duma, gathered for veches and, whatever the elders decided, the suburbs agreed to.

This article gives a starting point idea of ​​identities. Here we will define the identity, introduce the notation used, and, of course, give various examples identities

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What is identity?

It is logical to start presenting the material with identity definitions. In Makarychev Yu. N.’s textbook, algebra for 7th grade, the definition of identity is given as follows:

Definition.

Identity– this is an equality that is true for any values ​​of the variables; any true numerical equality is also an identity.

At the same time, the author immediately stipulates that this definition will be clarified in the future. This clarification occurs in 8th grade, after becoming familiar with the definition of permissible values ​​of variables and DL. The definition becomes:

Definition.

Identities- these are true numerical equalities, as well as equalities that are true for all permissible values ​​of the variables included in them.

So why, when defining identity, in 7th grade we talk about any values ​​of variables, and in 8th grade we start talking about the values ​​of variables from their DL? Until grade 8, work is carried out exclusively with whole expressions (in particular, with monomials and polynomials), and they make sense for any values ​​of the variables included in them. That’s why in 7th grade we say that identity is an equality that is true for any values ​​of the variables. And in the 8th grade, expressions appear that no longer make sense not for all values ​​of variables, but only for values ​​from their ODZ. Therefore, we begin to call equalities that are true for all admissible values ​​of the variables.

So identity is special case equality. That is, any identity is equality. But not every equality is an identity, but only an equality that is true for any values ​​of the variables from their range of permissible values.

Identity sign

It is known that in writing equalities, an equal sign of the form “=” is used, to the left and right of which there are some numbers or expressions. If we add another horizontal line to this sign, we get identity sign“≡”, or as it is also called equal sign.

The sign of identity is usually used only when it is necessary to especially emphasize that we are faced with not just equality, but identity. In other cases, records of identities do not differ in appearance from equalities.

Examples of identities

It's time to bring examples of identities. The definition of identity given in the first paragraph will help us with this.

Numerical equalities 2=2 are examples of identities, since these equalities are true, and any true numerical equality is by definition an identity. They can be written as 2≡2 and .

Numerical equalities of the form 2+3=5 and 7−1=2 3 are also identities, since these equalities are true. That is, 2+3≡5 and 7−1≡2·3.

Let's move on to examples of identities that contain not only numbers, but also variables.

Consider the equality 3·(x+1)=3·x+3. For any value of the variable x, the written equality is true due to distributive property multiplication relative to addition, therefore, the original equality is an example of identity. Here is another example of an identity: y·(x−1)≡(x−1)·x:x·y 2:y, here the range of permissible values ​​of the variables x and y consists of all pairs (x, y), where x and y are any numbers except zero.

But the equalities x+1=x−1 and a+2·b=b+2·a are not identities, since there are values ​​of the variables for which these equalities will not be true. For example, when x=2, the equality x+1=x−1 turns into the incorrect equality 2+1=2−1. Moreover, the equality x+1=x−1 is not achieved at all for any values ​​of the variable x. And the equality a+2·b=b+2·a will turn into an incorrect equality if we take any different values ​​of the variables a and b. For example, with a=0 and b=1 we will arrive at the incorrect equality 0+2·1=1+2·0. Equality |x|=x, where |x|

- variable x is also not an identity, since it is not true for negative values ​​of x.

Examples of the most well-known identities are of the form sin 2 α+cos 2 α=1 and a log a b =b.

In conclusion of this article, I would like to note that when studying mathematics we constantly encounter identities. Records of properties of actions with numbers are identities, for example, a+b=b+a, 1·a=a, 0·a=0 and a+(−a)=0. Also the identities are

Let's consider two equalities:

1. a 12 *a 3 = a 7 *a 8

This equality will hold for any values ​​of the variable a. The range of acceptable values ​​for that equality will be the entire set of real numbers.

2. a 12: a 3 = a 2 *a 7 . This inequality will be true for all values ​​of the variable a, except a. The range of acceptable values ​​for this inequality will be the entire set of real numbers except zero.

For each of these equalities it can be argued that it will be true for any admissible values ​​of the variables a. Such equalities in mathematics are called identities.

The concept of identity

An identity is an equality that is true for any admissible values ​​of the variables. If you substitute any valid values ​​into this equality instead of variables, you should get a correct numerical equality.

It is worth noting that true numerical equalities are also identities. Identities, for example, will be properties of actions on numbers.

3. a + b = b + a;

4. a + (b + c) = (a + b) + c;

5. a*b = b*a;

6. a*(b*c) = (a*b)*c;

7. a*(b + c) = a*b + a*c;

8. a + 0 = a;

9. a*0 = 0;

10. a*1 = a;

11. a*(-1) = -a.

If two expressions for any admissible variables are respectively equal, then such expressions are called identically equal. Below are some examples of identically equal expressions:

1. (a 2) 4 and a 8 ;

2. a*b*(-a^2*b) and -a 3 *b 2 ;

3. ((x 3 *x 8)/x) and x 10.

We can always replace one expression with any other expression identically equal to the first. Such a replacement will be an identity transformation.

Examples of identities

Example 1: are the following equalities identical:

1. a + 5 = 5 + a;

2. a*(-b) = -a*b;

3. 3*a*3*b = 9*a*b;

4. a-b = b-a.

Not all expressions presented above will be identities. Of these equalities, only 1, 2 and 3 equalities are identities. No matter what numbers we substitute in them, instead of variables a and b we will still get correct numerical equalities.

But 4 equality is no longer an identity. Because this equality will not hold for all valid values. For example, with the values ​​a = 5 and b = 2, the following result will be obtained:

5 - 2 = 2 - 5;

3 = -3.

This equality is not true, since the number 3 is not equal to the number -3.

LECTURE No. 3 Proof of identities

Purpose: 1. Repeat the definitions of identity and identically equal expressions.

2.Introduce the concept of identical transformation of expressions.

3. Multiplying a polynomial by a polynomial.

4. Factoring a polynomial using the grouping method.

Let every day and every hour

He'll get us something new,

Let our minds be good,

And the heart will be smart!

There are many concepts in mathematics. One of them is identity.

An identity is an equality that holds for all values ​​of the variables that are included in it. We already know some identities.

For example, everyone abbreviated multiplication formulas are identities.

Abbreviated multiplication formulas

1. (a ± b)2 = a 2 ± 2 ab + b 2,

2. (a ± b)3 = a 3 ± 3 a 2b + 3ab 2 ± b 3,

3. a 2 - b 2 = (a - b)(a + b),

4. a 3 ± b 3 = (a ± b)(a 2 ab + b 2).

Prove identity- this means establishing that for any valid variable value, its left side is equal to the right side.

There are several in algebra in various ways proofs of identities.

Methods for proving identities

Perform equivalent conversions left side of the identity. If we end up with the right-hand side, then the identity is considered proven. Perform equivalent conversions the right side of the identity. If we finally get the left side, then the identity is considered proven. Perform equivalent conversions left and right sides of the identity. If we get the same result, then the identity is considered proven. From the right side of the identity we subtract the left side. We perform equivalent transformations on the difference. And if in the end we get zero, then the identity is considered proven. The right side is subtracted from the left side of the identity. We perform equivalent transformations on the difference. And if in the end we get zero, then the identity is considered proven.

It should also be remembered that the identity is valid only for permissible values ​​of the variables.

As you can see, there are quite a lot of ways. Which method to choose in a given case depends on the identity you need to prove. As you prove various identities, you will gain experience in choosing a method of proof.

An identity is an equation that is satisfied identically, that is, valid for any admissible values ​​of the variables included in it. To prove an identity means to establish that for all admissible values ​​of the variables, its left and right sides are equal.
Ways to prove identity:
1. Perform transformations on the left side and ultimately obtain the right side.
2. Perform transformations on the right side and ultimately obtain the left side.
3. Separately transform the right and left sides and obtain the same expression in both the first and second cases.
4. Compose the difference between the left and right sides and, as a result of its transformations, obtain zero.
Let's look at some simple examples

Example 1. Prove the identity x·(a+b) + a·(b-x) = b·(a+x).

Solution.

Since the right side has a small expression, let's try to transform the left side of the equality.

x·(a+b) + a·(b-x) = x·a +x·b + a·b – a·x.

Let us present similar terms and take the common factor out of the bracket.

x a + x b + a b – a x = x b + a b = b (a + x).

We found that the left side after the transformations became the same as the right side. Therefore, this equality is an identity.

Example 2. Prove the identity: a² + 7·a + 10 = (a+5)·(a+2).

Solution:

IN in this example you can do it in the following way. Let's open the brackets on the right side of the equality.

(a+5)·(a+2) = (a²) + 5·a +2·a +10 = a²+7·a + 10.

We see that after the transformations, the right side of the equality became the same as the left side of the equality. Therefore, this equality is an identity.

“The replacement of one expression by another identically equal to it is called an identical transformation of the expression”

Find out which equality is an identity:

1. - (a – c) = - a – c;

2. 2 · (x + 4) = 2x – 4;

3. (x – 5) · (-3) = - 3x + 15.

4. рху (- р2 x2 y) = - р3 x3 y3.

“To prove that some equality is an identity, or, as they say differently, to prove an identity, identical transformations of expressions are used”

An equality that is true for any values ​​of the variables is called identity. To prove that some equality is an identity, or, as they say differently, so that prove identity, use identical transformations of expressions.
Let's prove the identity:
xy - 3y - 5x + 16 = (x - 3)(y - 5) + 1 Transform the left side of this equality:
xy - 3y - 5x + 16 = (xy - 3y) + (- 5x + 15) +1 = y(x - 3) - 5(x -3) +1 = (y - 5)(x - 3) + 1 As a result identity transformation from the left side of the polynomial we obtained its right side and thereby proved that this equality is identity.
For proofs of identity transform its left side into the right or its right side into the left, or show that the left and right sides of the original equality are identically equal to the same expression.

Multiplying a polynomial by a polynomial

Let's multiply the polynomial a+b to a polynomial c + d. Let's compose the product of these polynomials:
(a+b)(c+d).
Let us denote the binomial a+b letter x and transform the resulting product according to the rule for multiplying a monomial by a polynomial:
(a+b)(c+d) = x(c+d) = xc + xd.
In expression xc + xd. let's substitute x polynomial a+b and again use the rule for multiplying a monomial by a polynomial:
xc + xd = (a+b)c + (a+b)d = ac + bc + ad + bd.
So: (a+b)(c+d) = ac + bc + ad + bd.
Product of polynomials a+b And c + d we represented it as a polynomial ac + bc + ad + bd. This polynomial is the sum of all monomials resulting from multiplying each term of the polynomial a+b for each term of the polynomial c + d.
Conclusion: the product of any two polynomials can be represented as a polynomial.
Rule: To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of another polynomial and add the resulting products.
Note that when multiplying a polynomial containing m terms to a polynomial containing n terms in the product, before bringing similar terms, the result should be mn members. This can be used for control.

Factoring a polynomial using the grouping method:

Previously, we were introduced to factoring a polynomial by taking the common factor out of brackets. Sometimes it is possible to factor a polynomial using another method - grouping of its members.
Let's factor the polynomial
ab - 2b + 3a - 6 Let's group it so that the terms in each group have a common factor and take this factor out of brackets:
ab - 2b + 3a - 6 = (ab - 2b) + (3a - 6) = b(a - 2) + 3(a - 2) Each term of the resulting expression has a common factor (a - 2). Let's take this common factor out of brackets:
b(a - 2) + 3(a - 2) = (b +3)(a - 2) As a result, we factored the original polynomial:
ab - 2b + 3a - 6 = (b +3)(a - 2) The method we used to factor the polynomial is called grouping method.
Polynomial expansion ab - 2b + 3a - 6 factorization can be done by grouping its terms differently:
ab - 2b + 3a - 6 = (ab + 3a) + (- 2b - 6) = a(b + 3) -2(b + 3) = (a - 2)(b + 3)

Repeat:

1. Methods of proving identities.

2. What is called the identity transformation of an expression.

3. Multiplying a polynomial by a polynomial.

4. Factoring a polynomial using the grouping method