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Identities. Identity transformations of expressions. 7th grade.
Find the value of the expressions at x=5 and y=4 3(x+y)= 3(5+4)=3*9=27 3x+3y= 3*5+3*4=27 Find the value of the expressions at x=6 and y=5 3(x+y)= 3(6+5)=3*11=33 3x+3y= 3*6+3*5=33
CONCLUSION: We got the same result. It follows from the distributive property that, in general, for any values of the variables, the values of the expressions 3(x + y) and 3x + 3y are equal. 3(x+y) = 3x+3y
Consider now the expressions 2x + y and 2xy. for x=1 and y=2 they take equal values: 2x+y=2*1+2=4 2x=2*1*2=4 for x=3, y=4 expression values are different 2x+y=2* 3+4=10 2xy=2*3*4=24
CONCLUSION: The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal. Definition: Two expressions whose values are equal for any values of the variables are said to be identically equal.
IDENTITY The equality 3(x+y) and 3x+3y is true for any values of x and y. Such equalities are called identities. Definition: An equality that is true for any values of the variables is called an identity. True numerical equalities are also considered identities. We have already met with identities.
Identities are equalities expressing the basic properties of actions on numbers. a + b = b + a ab = ba (a + b) + c = a + (b + c) (ab)c = a(bc) a(b + c) = ab + ac
Other examples of identities can be given: a + 0 = a a * 1 = a a + (-a) = 0 a * (- b) = - ab a- b = a + (- b) (-a) * ( -b) = ab Replacing one expression with another expression identically equal to it is called identity transformation or simply expression transformation.
To bring like terms, you need to add their coefficients and multiply the result by the common letter part. Example 1. We give like terms 5x + 2x-3x \u003d x (5 + 2-3) \u003d 4x
If there is a plus sign in front of the brackets, then the brackets can be omitted, preserving the sign of each term enclosed in brackets. Example 2. Expand the brackets in the expression 2a + (b -3 c) = 2 a + b - 3 c
If there is a minus sign before the brackets, then the brackets can be omitted by changing the sign of each term enclosed in brackets. Example 3. Let's open the brackets in the expression a - (4 b - c) \u003d a - 4 b + c
Homework: p. 5, No. 91, 97, 99 Thank you for the lesson!
On the topic: methodological developments, presentations and notes
Methods of preparing students for the exam in the section "Expressions and transformation of expressions"
This project was developed with the aim of preparing students for state exams in grade 9 and later on for a unified state exam in grade 11....
So, friends, in the last lesson we met with Understand what the words mean "expression doesn't make sense". And now it's time to figure it out What is expression transformation? And the most important thing - why is it needed.
What is Expression Transformation?
The answer is simple, obscenely.) This any action with an expression. And that's it. You have been doing all these transformations since the first class. Any not literally, of course ... This will be discussed below.)
For example, let's take some super-cool numerical expression Let's say 3+2. How can it be converted? Yes, very easy! At least take and count:
3+2 = 5
This calculation of the kindergarten will be expression conversion. You can write the same expression in a different way:
3+2 = 2+3
And here we did not count anything at all. Just took and rewrote our expression in a different form. it will also be a transformation of the expression. It can also be written differently. For example, like this:
3+2 = 10-5
And this entry is also a transformation of an expression.
Or like this:
3+2 = 10:2
Also transformation of the expression!
If you and I are older, we are friends with algebra, then we will write:
Whoever is on "you" with algebra, even without particularly straining and not counting anything, will realize in his mind that there is an ordinary five on the left and on the right. Tighten up and try.)
And if we are really older, then we can write down such horror films:
log 2 8+ log 2 4 = log 2 32
Or even like this:
5 sin 2 x+5 cos 2 x=5 tgx ctgx
Does it inspire? And such transformations, obviously, can be done as much as you want! As far as fantasy allows. And a set of knowledge of mathematics.)
Did you get the point?
Any action on an expression any writing it in another form is called expression conversion. And all things. Everything is very simple.
Simplicity, of course, is always a good and pleasant thing, but you have to pay somewhere for any simplicity, yes .... There is one major "but" here. All these mysterious transformations always obey one very important rule. This rule is so important that it can safely be called main rule all mathematics. And breaking that simple rule inevitably will lead to errors. Do we understand?)
Suppose we have transformed our expression haphazardly, from a bullshit, somehow like this:
3+2 = 6+1
Transformation? Of course. We wrote the expression in a different form! But... what's wrong here?
Answer: everything is not so.) The thing is that the transformations "anyhow andfrom the bulldozer" Mathematics is not interested at all.) Why? Because all mathematics is built on transformations in which the appearance changes, but the essence of the expression does not change. This is her strict requirement. And violation of this requirement will lead to errors. Three plus two can be written in any form. In which example it requires, we will write it in that form. But inherently this is should always be five. In whatever form we write down these same 3 + 2. But, if, suddenly, after writing the expression 3 + 2 in a different form, instead of five, you will have twenty five, somewhere you made a mistake along the way. Come back and fix it.)
And now it's time for wise green thoughts.)
Remember:
1. Any action on an expression, writing it in a different form, is called expression transformation.
2. Transformations,expressions that do not change the essence, are called identical.
3. All mathematics is built on identical transformations of expressions.
Exactly identical transformations and allow us, step by step, little by little, to turn a complex example into a simple, white and fluffy expression, while maintaining essence of the example. If, suddenly, in the chain of our transformations we make a mistake somewhere, and at some step we make a NOT IDENTICAL transformation, then we will decide further completely different example. With other answers, yes ... Which will no longer have anything to do with the correct ones.) Let's break the identity and mess up somewhere else - let's start solving it already third example. And so on, depending on the number of stocks, from the problem about the train and the car, you can come to the problem about one and a half excavators.)
Another example. For students who are already studying algebra with might and main. Let's say we need to find the value of the expression (40+7) 2 . How can you get out, i.e. transform our evil expression? You can simply calculate the expression in brackets (we get 47), multiply by a column by itself and get (if you count) 2209. Or you can use the formula
(a+b) 2 = a 2 +2ab+b 2 .
We get: (40+7) 2 = 40 2 +2∙40∙7+7 2 = 1600+560+49 = 2209.
But! There is a temptation (say, due to ignorance of the formula) when squaring to write simply:
(40+7) 2 = 40 2 +7 2 .
Unfortunately, on this simple and seemingly obvious transition, the identity of our transformations violated. On the left, everything is as it should, 2209, but on the right - already another number. 1649. Calculate - and everything will become clear. Here is a typical example of a NOT identical transformation. And accordingly got out errors.)
Here it is and the main rule for solving any tasks: compliance with the identity of the transformations.
I gave an example with numerical expressions 3 + 2 and (40 + 7) 2 purely for clarity.
What about algebraic expressions? All the same! Only in algebraic expressions identical transformations are given formulas and rules. Let's say there is a formula in algebra:
a(b-c) = ab - ac
So, in any example, we have every right to replace the expression a(b-c) feel free to write an alternative expression ab-ac. And vice versa. This Mathematics provides us with a choice of these two expressions. And which one to write depends on the specific example.
Or popular:
a 2 - b 2 = (a- b)(a+ b)
Again, there are two possibilities. Both are correct.) This too identical transformation. What is more profitable to write - the difference of squares or the product of brackets - an example will tell you.)
Another example. One of the most important and necessary transformations in mathematics is basic property of a fraction. You can (will) read and watch the link for more details (when the lesson is done), but here I’ll just remind the rule:
If the numerator and denominator of a fraction are multiplied (divided) by samenumber, or an expression that is not equal to zero, the fraction will not change.
Here is an example of identical transformations for this property:
As you probably guessed, this glorious chain can be continued indefinitely ...) As long as the creative impulse is enough. All sorts of minuses, roots, don't let them confuse you. It's all same fraction. By its essence. Two-thirds. 2/3. Just written in different forms.:) Very important property. It is it that very often allows you to turn all sorts of example monsters into white and fluffy.)
Of course, there are many formulas and rules that define identical transformations. I would even say a lot. But the most important ones, without which in mathematics at least the triple level can be dispensed with it is forbidden, is a perfectly reasonable amount.
Here are some of the basic transformations:
1. Working with monomials and polynomials. Reduction of similar terms (or shortly - similar);
2. Parentheses opening and parenthesis enclosing ;
3. Factorization ;
4. and decomposition of a square trinomial.
5. Working with fractions and fractional expressions.
These five basic transformations are widely used in all mathematics. From elementary to advanced. And, if you do not own at least one of these five simple things, then you will inevitably face big problems both in all of mathematics in high school and in high school, and even more so at the university. Therefore, we will start with them. in later lessons in this section.)
There are even cooler transformations. For advanced schoolchildren and students.) Be it:
6., and everything connected with them;
7. Full square selection from a square trinomial;
8. Division of polynomials corner or according to Horner's scheme ;
9. Decomposition of a rational fraction into a sum of elementary (simple) fractions. The most useful feature for students at work
So, everything is clear about the identity of the transformations and the importance of observing it? Excellent! Then it's time to move to the next level and step from primitive arithmetic to more serious algebra finally. And with a twinkle in his eye.)
Identity transformations
1. The concept of identity. Main types of identical transformations and stages of their study.
The study of various transformations of expressions and formulas occupies the smallest part of the study time in the course of school mathematics. The simplest ^ "" formations, based on the properties of arithmetic operations, are already in elementary school. But the main burden on the formation of skills and abilities to perform transformations is borne by the course of school algebra 1> this is connected:
with a sharp increase in the number of transformations performed, their diversity;
with the complication of activities to substantiate them and clarify the conditions of applicability;
i) with the allocation and study of generalized concepts of identity, identical transformation, equivalent transformation, logical consequence.
The line of identical transformations receives the following development in the algebra course of the basic school:
,4 b classes - opening the brackets, bringing like terms, take out - M (Chsho factor out of the brackets;
7 Class - identical transformations of integer and fractional expressions;
H class - identical transformations of expressions containing quad roots;
( > class- identical transformations of trigonometric expressions and mmrizhsny containing a degree with a rational exponent.
The line of identical transformations is one of the important ideological lines of the algebra course. Therefore, teaching mathematics in grades 5-6 is built by niKiiM in such a way that students already in these grades acquire the skills of the simplest identical transformations (without using the term “identically different transformations”). These skills are formed when performing an exercise to reduce similar terms, open brackets and put them in brackets, put a factor out of brackets, etc. The simplest transformations of numeric and alphabetic expressions are also considered. At this level of learning, transformations are mastered that are performed directly on the basis of the laws and properties of arithmetic operations.
The main types of tasks in grades 5-6, in the solution of which the properties and laws of arithmetic operations are actively used and through which the skills of identical transformations are formed, include:
substantiation of algorithms for performing actions on the numbers of the studied numerical sets;
calculating the values of a numeric expression in the most rational way;
comparing the values of numeric expressions without performing the specified actions;
simplification of literal expressions;
proof of the equality of the values of two literal expressions, etc.
Express the number 153 as a sum of bit terms; as a difference of two numbers, as a product of two numbers.
Express the number 27 as the product of three identical factors.
These exercises on the representation of the same number in different forms of notation contribute to the assimilation of the concept of identical transformations. Initially, these representations can be arbitrary, in the future - purposeful. For example, the representation as a sum of bit terms is used to explain the rules for adding natural numbers in a “column”, the representation as a sum or difference of “convenient” numbers is used to perform quick calculations of various products, and the representation as a product of factors is used to simplify various fractional expressions.
Find the value of the expression 928 36 + 72 36.
The rational way to calculate the value of this expression is based on the use of the distributive law of multiplication with respect to addition: 928 36 + 72 36 = (928 + 72) 36 = 1000 36 = 36000.
In the school course of mathematics, the following stages of mastering the application of transformations of alphanumeric expressions and formulas can be distinguished.
stage. Beginnings of algebra. At this stage, an undivided system of transformations is used; it is represented by the rules for performing actions on one or both parts of the formula.
Example. Solve Equations:
a) 5x - bx = 2; b) 5x = 3x + 2; in) 6 (2 - 4y) + 5u = 3 (1 - Zu).
The general idea of the solution is to simplify these formulas with the help of several rules. In the first task simplification is achieved by applying the identity: 5x- bx= (5 - 3)x. The identity transformation based on this identity transforms the given equation into an equivalent urshshomie 2x - 2.
Second equation requires for its solution not only an identical, but also a paranoid transformation; as such, it uses the right ||n to transfer the terms of the equation from one part of the equation to another with a modified chic. In solving such a simple task as b), both mon in transformations are used - both identical and equivalent. This proposition is also used for more cumbersome tasks, such as the third one.
The mole of the first stage is to teach how to quickly solve the simplest equations, simplify the formulas that define functions, rationally carry out calculations based on the properties of actions.
tit. Formation of skills for the application of specific types of transformationII tilt The concepts of identity and identical transformation are explicitly introduced in the course shn "sbra 7 class. So, for example, in Yu. N. Makarychev's textbook "Algebra 7" np" the concept of identically equal expressions is introduced: variables, spy identically equal" then the concept of identity: “An equality that is paired for any values of the variables is called identity."
11 examples are given:
In the textbook A.G. Mordkovich "Algebra 7" is given immediately and a refined concept of identity: "Identity is the correct equality for any allowable values of its constituent variables.
When introducing the concept of an identical transformation, one should first of all dismiss the expediency of studying identical transformations. To do this, you can consider various exercises to find the meaning of expressions.
liiiipiiMep, find the value of the expression 37.1x + 37,ly when X= 0.98, y = 0.02. Using the distributive property of multiplication, the expression 37.1l + 37.1 at can be calculated by the expression 37.1(x + y), identically equal to him. Even more impressive worm 1 solution of the following exercise: find the value of the expression
() - (a-6) _ p r i. a) q = z > ^ = 2; b) a = 121, b - 38; c) a = 2.52, b= 1 -.
ab 9
After the transformations carried out, it turns out that the set of values of this expression consists of a single number 4.
In Yu. N. Makarychev's textbook “Algebra 7”, the introduction of the concept of an identity transformation is motivated by considering an example: “To find the value of the expression xy-da for x = 2.3; y = 0.8; z = 0.2, you need to perform 3 actions: hu - xz = 2,3 0,8 - 2,3 0,2 = 1,84 - 0,46 = 1,38.
11 It is necessary to note one type of transformations, specific for Curonian algebra and the beginnings of analysis. These are transformations of expressions containing pre-transitions, and transformations based on the rules of differentiation and integration. The main difference between these "analytic" transformations and "algebraic" transformations lies in the nature of the set that the variables run through in the identities. In algebraic identities, the variables run through number areas and in the analytic sets these sets wander certain many functions. For example, the rule of differentiating the sum: (Z "+g)" here / and g are variables running through the sets
I I but differentiable functions with a common domain of definition. Outwardly, these transformations are similar to transformations of an algebraic type, therefore, sometimes they say "algebra of limits", "algebra of differentiation".
The identities studied in the school course of algebra and the algebraic mayrial of the course of algebra and the beginnings of analysis can be divided into two classes.
The first one consists of the identities of the reduced multiplication, fair in
av v.
iioGom commutative ring, and the identities - =-,a* 0, fair in any
Om field.
The second class is formed by identities connecting arithmetic expressions and basic elementary functions, as well as compositions of elementaryHhixfunctions. Most of the identities of this class also have a common mathematical basis, consisting in the fact that the power, exponential and logarithmic functions are isomorphisms of various numerical groups. For example, there is a statement: there is a unique continuous isomorphic mapping / of the additive group of real numbers into the multiplicative group of positive real numbers, under which the unit o is mapped to a given number a> 0, a f 1; this mapping is given by a reciprocal function with base a:/(X)= a. There are similar assertions for power and logarithmic functions.
The methodology for studying the identities of both classes has many common features. In general, the identity transformations studied in the school mathematics course include:
transformations of expressions containing radicals and degrees with fractional exponents;
transformations of expressions containing passages to the limit, and transformations based on the rules of differentiation and integration.
This result can be obtained by performing only two actions, using the expression x (y-z), identically equal to the expression xy-xz:x(y-z)= 2,3 (0,8 - 0,2) = 2,3 0,6 = 1,38.
We have simplified the calculations by replacing the expression xy-xz identically equal expression x (y - z).
The replacement of one expression by another, identically equal to it, is called identity transformation or simply expression transformation.
The development of various types of transformations at this stage begins with the introduction of abbreviated multiplication formulas. Then we consider transformations associated with the operation of raising to a power, with various classes of elementary functions - exponential, power, logarithmic, trigonometric. Each of these types of transformations goes through a stage of study, in which attention is focused on the assimilation of their characteristic features.
With the accumulation of material, it becomes possible to single out and, on this basis, introduce the concepts of identical and equivalent transformations.
It should be noted that the concept of an identical transformation is given in the school course of algebra not in full generality, but only in application to expressions. Transformations are divided into two classes: identical transformations are expression transformations, and equivalent - formula transformations. In the case when there is a need to simplify one part of the formula, an expression is highlighted in this formula, which serves as an argument for the applied identical transformation. For example, the equations 5x - Zx - 2 and 2x = 2 are considered not only equivalent, but the same.
In the textbooks of algebra Sh.A. Alimova and others, the concept of identity is not explicitly introduced in grades 7-8 and only in grade 9 in the topic “Trigonometric identities” when solving problem 1: “Prove that for afkk, to < eZ , the equality 1 + ctg 2 a = -\-» is valid, this concept is introduced. Here it is explained to the students that sin a
the indicated equality is “valid for all admissible values of a, i.e. such that its left and right parts make sense. Such equalities are called identities, and problems for proving such equalities are called problems for proving identities.
III stage. Organization of an integral system of transformations (synthesis).
The main goal of this stage is to form a flexible and powerful apparatus suitable for use in solving a variety of educational tasks.
The deployment of the second stage of the study of transformations takes place throughout the entire course of algebra in the basic school. The transition to the third stage is carried out during the final repetition of the course in the course of comprehending the already known material, learned in parts, according to certain types of transformations.
In the course of algebra and the beginning of analysis, an integral system of transformations, basically already formed, continues to be gradually improved. Some new types of transformations are also added to it (for example, those related to trigonometric and logarithmic functions), but they only enrich it, expand its capabilities, but do not change its structure.
The methodology for studying these new transformations practically does not differ from that used in the course of algebra.
It is necessary to note one type of transformations specific to the courses of algebra and the beginnings of analysis. These are transformations of expressions containing limit transitions, and transformations based on the rules of differentiation and integration. The main difference between these "analytic" transformations and "algebraic" transformations lies in the character of the set, which the variables run through in the identities. In algebraic identities, the variables run through number areas while in analytic sets certain many functions. For example, the sum differentiation rule: ( f + g )" = f + g "; here fug - variables running through multiplier differentiable functions with a common domain of definition. Outwardly, these transformations are similar to transformations of an algebraic type, therefore, sometimes they say "algebra of limits", "algebra of differentiation".
The identities studied in the school course of algebra and the algebraic material of the course of algebra and the beginnings of analysis can be divided into two classes.
The first one consists of the identities of the reduced multiplication, fair in
any commutative ring, and the identity - \u003d -, a * 0, valid in any
ac s
The second class is formed by identities connecting arithmetic operations and basic elementary functions, as well as compositions of elementary functions. Most of the identities of this class also have a common mathematical basis, which consists in the fact that the power, exponential and logarithmic functions are isomorphisms of various numerical groups. For example, there is a statement: there is a unique continuous isomorphic mapping / of the additive group of real numbers into the multiplicative group of positive real numbers, under which one is mapped to a given number a> 0, a f one; this mapping is given by the exponential function with base i: / (x) = a*. There are similar assertions for power and logarithmic functions.
The methodology for studying the identities of both classes has many common features. In general, the identity transformations studied in the school mathematics course include:
transformations of algebraic expressions;
conversion of expressions containing radicals and powers with fractional exponents;
transformations of trigonometric expressions;
conversion of expressions containing powers and logarithms;
transformations of expressions containing limit transitions, and transformations based on rules, differentiation and integration.
2. Features of the organization of the task system in the study of identical transformations
The basic principle of organizing any system of tasks is to present them from simple to complex taking into account the need for students to overcome feasible difficulties and create problem situations. This basic principle requires concretization in relation to the features of this educational material. Let's give an example of a system of exercises on the topic: "The square of the sum and
difference of two numbers.
I la this basic system of exercises ends. Such a system should ensure the assimilation of the basic material.
The following exercises (17-19) allow students to focus on typical mistakes and contribute to the development of interest and their creative abilities.
In each case, the number of exercises in the system may be less or more, but the sequence of their implementation should be the same.
To describe various task systems in the methodology of mathematics, use also the concept exercise cycle. The cycle of exercises is characterized by the fact that several aspects of the study and methods of arranging the material are combined into a sequence of exercises. In relation to identical transformations, the idea of a cycle can be given as follows.
A cycle of exercises is connected with the study of one identity, around which other identities are grouped, which are in a natural connection with it. The "stop the loop along with executive includes tasks that require recognize-< ii in nor the applicability of the considered identity. The identity under study is used to perform calculations on various numerical domains.
Tasks in each cycle are divided into two groups. To first include tasks that are completed during the initial acquaintance with the identity. They are performed in several lessons united by one topic. Second group exercise connects the identity under study with various applications. The exercises in this group are usually scattered around different topics.
The described structure of the cycle refers to the stage of formation of skills for applying specific types of transformations. At the final stage - (Thane of synthesis, the cycles are modified. Firstly, both groups of shdapia are combined, forming "unrolled" cycle , and write down the simplest in terms of wording or complexity of execution from the first group. The remaining types of tasks become more difficult. Secondly, there is a merging of cycles related to different identities, because of this, the role of actions to recognize the applicability of one or another identity increases.
Let's take a concrete example of a loop.
Example. Job cycle for identity x -y 2 = (x-y)(x + y).
The execution of the first group of tasks of this cycle occurs as follows:
conditions. The students have just got acquainted with the formulation of the identity (or rather, with two formulations: “The difference of the squares of two expressions is equal to the product of the sum and difference of these expressions” and “The product of the sum and difference of two expressions is equal to the difference of the squares of these expressions”), its writing in the form of a formula, proof . Following this, several examples of the use of a transformation based on this identity are given. Finally, the students begin to do the exercises on their own.
First group of tasks
The second group of tasks
(The tasks of each group can be presented to students using a multimedia projector)
Let us carry out a methodical analysis of this system of task types.
The task a0 aims to fix the structure of the identity under study. This is achieved by replacing the letters (x and y) in the notation of identity in other letters. Tasks of this type allow you to clarify the relationship between the verbal expression and the symbolic form of identity.
Task a 2) is focused on establishing the connection of this identity with the numerical system. The expression being converted here is not purely literal, but alphanumeric. To describe the actions performed, it is necessary to use the concept substitution letters by number in identity. Skill Development
the use of the substitution operation and the deepening of the idea of it is carried out in the performance of tasks of the type r 2).
The next step in mastering the identity is illustrated by task a). In the new task, the expression proposed for transformation does not have the form of a split of squares; transformation becomes possible only when. h(n1k will notice that the number 121 can be represented as a square of a number. Thus, this task is performed not in one step, but in two: on the laneiiiu there is a recognition of the possibility of reducing this expression to the mdf of the difference of squares, on the second a transformation is performed using the identity.
During the first stages of mastering the identity, each step is recorded:
I "I / s 2 \u003d 11 2 - & 2 \u003d (11 - £) (11 + to), in the future, some recognition operations are performed by students orally.
In example dd) it is required to establish connections between this identity and others related to actions with monomials; e 3) one should apply the identity for the difference of squares twice; c) students will have to overcome a certain psychological barrier, exercising access to the area of irrational numbers.
Tasks of type b) are aimed at developing the skills of replacing the product (,v - y)(x + y) to the difference X 2 - at 2 . Tasks of type c) play a similar role. In examples of type d) it is required to choose one of the directions of transformations.
In general, the tasks of the first group are focused on mastering the structure of identity, substitution operations in the simplest most important cases, and ideas about the reversibility of transformations carried out by identity,
The main features and objectives disclosed by us when considering the first | the ruins of the tasks of the cycle, refer to any cycle of exercises that form the bayonets of the use of identity. For any newly introduced peri-im identity, the group of tasks in the cycle must retain the features described here; the only difference is in the number of tasks.
1 The second group of tasks in the cycle, unlike the first, is aimed at the fullest possible use and consideration of the specifics of this particular identity t i pi. The tasks of this group assume already formed skills of using the identity for the difference of squares (in the simplest cases); cpi, the tasks of this group are to deepen the understanding of identity by considering its various applications in various situations, in combination with the use of material related to other topics of the mathematics course.
Consider the solution of task l):
x 3 - 4x \u003d 15 o x 3 - 9x \u003d 15 - 5x o x (x ~ 3) (x + 3) \u003d 5 (3 -x) ox \u003d 3, or \{\ 1-3) = -5. The equation x(x + 3) = -5 has no real roots, therefore \ 3 is the only root of the equation.
We see that the use of the identity for the difference of squares is part of the n and I part in solving the example, being the leading idea for carrying out transformations.
Task cycles associated with identities for elementary functions have their own characteristics, which are due to the fact that, firstly. the corresponding identities are studied in connection with the study of functional material and, /u>-“touykh, they appear later than the identities of the first group and are studied with
using already formed skills for carrying out identical transformations. A significant part of the use of identity transformations associated with elementary functions falls on the solution of irrational and transcendental equations. The cycles related to the assimilation of identities include only the simplest equations, but already here it is advisable to carry out work on mastering the method of solving such equations: reducing it by replacing the unknown to an algebraic equation.
The sequence of steps for this solution is as follows:
a) find a function<р, для которой данное уравнение/(х) = 0 представимо в виде F (ср(лг)) = 0;
b) make a substitution at= cp(x) and solve the equation F(y) = 0;
c) solve each of the equations <р(х) = where (at k ) is the set of roots of the equation F(y) = 0.
A new issue that must be taken into account when studying identities with elementary functions is the consideration of the domain of definition. Here are examples of three tasks:
a) Plot the function y \u003d 4 log 2 x.
b) Solve the lg equation X + lg (x - 3) = 1.
c) On what set is the formula lg (x - 5) + lg (x + 5) = lg ( X 2 - 25) is an identity?
A typical mistake that students make in solving task a) is to use the equality a 1st excluding condition b > 0. In this case, as a result, the desired graph turns out to have the form of a parabola instead of the correct answer - the right branch of the parabola. In task b) one of the sources for obtaining complex systems of equations and inequalities is shown, when it is necessary to take into account the domains of definition of functions, and in task c) - an exercise that can serve as a preparatory exercise.
The idea that unites these tasks - the need to study the domain of definition of a function, can only be revealed by comparing such tasks that are heterogeneous in external form. The significance of this idea for mathematics is very great. It can serve as the basis for several cycles of exercises - for each of the classes of elementary functions.
In conclusion, we note that the study of identical transformations in school is of great importance. educational value. The ability to make some calculations, carry out calculations, for a long time with unremitting attention to follow some object is necessary for people of a wide variety of professions, regardless of whether they work in the field of mental or physical labor. The specificity of the section "Identical transformations of expressions" is such that it opens up wide opportunities for developing these important professionally significant skills in students.
Along with the study of operations and their properties in algebra, they study such concepts as expression, equation, inequality . The initial acquaintance with them occurs in the initial course of mathematics. They are introduced, as a rule, without strict definitions, most often ostensively, which requires the teacher not only to be very careful in the use of terms denoting these concepts, but also to know a number of their properties. Therefore, the main task that we set when starting to study the material of this paragraph is to clarify and deepen knowledge about expressions (numerical and with variables), numerical equalities and numerical inequalities, equations and inequalities.
The study of these concepts is associated with the use of a mathematical language, it refers to artificial languages that are created and developed along with a particular science. Like any other mathematical language, it has its own alphabet. In our course, it will be presented partially, due to the need to pay more attention to the relationship between algebra and arithmetic. This alphabet includes:
1) numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; with their help, numbers are written according to special rules;
2) signs of operations +, -, , :;
3) relationship signs<, >, =, M;
4) lowercase letters of the Latin alphabet, they are used to designate numbers;
5) brackets (round, curly, etc.), they are called technical signs.
Using this alphabet, words are formed in algebra, calling them expressions, and sentences are obtained from words - numerical equalities, numerical inequalities, equations, inequalities with variables.
As you know, records 3 + 7, 24: 8, 3 × 2 - 4, (25 + 3)× 2-17 are called numerical expressions. They are formed from numbers, action signs, brackets. If we perform all the actions indicated in the expression, we get a number called the value of a numeric expression . So, the value of the numeric expression is 3 × 2 - 4 is equal to 2.
There are numeric expressions whose values cannot be found. Such expressions are said to be don't make sense .
For example, expression 8: (4 - 4) does not make sense, since its value cannot be found: 4 - 4 = 0, and division by zero is impossible. The expression 7-9 also does not make sense if we consider it on the set of natural numbers, since the values of the expression 7-9 cannot be found on this set.
Consider the notation 2a + 3. It is formed from numbers, action signs and the letter a. If instead of a we substitute numbers, then different numerical expressions will be obtained:
if a = 7, then 2 × 7 + 3;
if a = 0, then 2 × 0 + 3;
if a = - 4, then 2 × (- 4) + 3.
In the notation 2a + 3, such a letter a is called variable , and the entry itself 2a + 3 - variable expression.
A variable in mathematics, as a rule, is denoted by any lowercase letter of the Latin alphabet. In elementary school, other characters are used to denote a variable in addition to letters, such as . Then the expression with a variable has the form: 2× + 3.
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 expression scope .
For example, the domain of the expression 5: (x - 7) consists of all real numbers, except for the number 7, since for x = 7 the expression 5: (7 - 7) has no meaning.
In mathematics, expressions are considered that contain one, two or more variables.
For example, 2a + 3 is a one-variable expression, and (3x + 8y) × 2 is an expression with three variables. In order to obtain a numerical expression from an expression with three variables, instead of each variable, substitute the numbers that belong to the scope of the expression.
So, we have found out how numerical expressions and expressions with variables are formed from the alphabet of the mathematical language. If we draw an analogy with the Russian language, then expressions are the words of the mathematical language.
But, using the alphabet of the mathematical language, it is possible to form such, for example, records: (3 + 2)) - × 12 or 3x - y: +) 8, which cannot be called either a numeric expression or an expression with a variable. These examples indicate that the description - from which characters of the alphabet of the mathematical language expressions are formed, numerical and with variables, is not a definition of these concepts. Let's give a definition of a numeric expression (an expression with variables is defined similarly).
Definition.If f and q are numeric expressions, then (f) + (q), (f) - (q), (f) × (q), (f) (q) are numerical expressions. Each number is considered to be a numeric expression.
If this definition were followed exactly, then one would have to write too many brackets, for example, (7) + (5) or (6): (2). To shorten the notation, we agreed not to write brackets if several expressions are added or subtracted, and these operations are performed from left to right. In the same way, brackets are not written when several numbers are multiplied or divided, and these operations are performed in order from left to right.
For example, they write like this: 37 - 12 + 62 - 17 + 13 or 120:15-7:12.
In addition, we agreed to first perform the actions of the second stage (multiplication and division), and then the actions of the first stage (addition and subtraction). Therefore, the expression (12-4:3) + (5-8:2-7) is written as follows: 12 - 4: 3 + 5 - 8: 2 - 7.
A task. Find the value of the expression 3x (x - 2) + 4(x - 2) for x = 6.
Solution
1 way. Substitute the number 6 instead of a variable in this expression: 3 × 6-(6 - 2) + 4 × (6 - 2). To find the value of the resulting numerical expression, we perform all the indicated actions: 3 × 6 × (6 - 2) + 4 × (6-2) = 18 × 4 + 4 × 4 = 72 + 16 = 88. Therefore, when X= 6 the value of the expression 3x(x-2) + 4(x-2) is 88.
2 way. Before substituting the number 6 in this expression, let's simplify it: Zx (x - 2) + 4 (x - 2) = (X - 2)(3x + 4). And then, substituting in the resulting expression instead of X number 6, do the following: (6 - 2) × (3 × 6 + 4) = 4x (18 + 4) = 4x22 = 88.
Let us pay attention to the following: both in the first method of solving the problem, and in the second one, we replaced one expression with another.
For example, the expression 18 × 4 + 4 × 4 was replaced by the expression 72 + 16, and the expression 3x (x - 2) + 4(x - 2) - by the expression (X - 2)(3x + 4), and these substitutions lead to the same result. In mathematics, describing the solution of this problem, they say that we performed identical transformations expressions.
Definition.Two expressions are said to be identically equal if, for any values of the variables from the domain of the expressions, their corresponding values are equal.
Examples of identically equal expressions are the expressions 5(x + 2) and 5x+ 10, because for any real values X their values are equal.
If two expressions that are identically equal on a certain set are joined by an equal sign, then we get a sentence called identity on this set.
For example, 5(x + 2) = 5x + 10 is an identity on the set of real numbers, because for all real numbers the values of the expression 5(x + 2) and 5x + 10 are the same. Using the general quantifier notation, this identity can be written as follows: (" x н R) 5(x + 2) = 5x + 10. True numerical equalities are also considered identities.
Replacing an expression with another that is identically equal to it on some set is called the identical transformation of the given expression on this set.
So, replacing the expression 5(x + 2) with the expression 5x + 10, which is identically equal to it, we performed the identical transformation of the first expression. But how, given two expressions, to find out whether they are identically equal or not? Find the corresponding values of expressions by substituting specific numbers for variables? Long and not always possible. But then what are the rules that must be followed when performing identical transformations of expressions? There are many of these rules, among them are the properties of algebraic operations.
A task. Factor the expression ax - bx + ab - b 2 .
Solution. Let's group the members of this expression in two (the first with the second, the third with the fourth): ax - bx + ab - b 2 \u003d (ax-bx) + (ab-b 2). This transformation is possible based on the associativity property of addition of real numbers.
We take out the common factor in the resulting expression from each bracket: (ax - bx) + (ab - b 2) \u003d x (a - b) + b (a - b) - this transformation is possible based on the distributive property of multiplication with respect to the subtraction of real numbers.
In the resulting expression, the terms have a common factor, we take it out of brackets: x (a - b) + b (a - b) \u003d (a - b) (x - b). The basis of the performed transformation is the distributive property of multiplication with respect to addition.
So, ax - bx + ab - b 2 \u003d (a - b) (x - b).
In the initial course of mathematics, as a rule, only identical transformations of numerical expressions are performed. The theoretical basis of such transformations are the properties of addition and multiplication, various rules: adding a sum to a number, a number to a sum, subtracting a number from a sum, etc.
For example, to find the product of 35 × 4, you need to perform transformations: 35 × 4 = (30 + 5) × 4 = 30 × 4 + 5 × 4 = 120 + 20 = 140. The performed transformations are based on: the distributive property of multiplication with respect to addition; the principle of writing numbers in the decimal number system (35 = 30 + 5); rules for multiplication and addition of natural numbers.
Let two algebraic expressions be given:
Let's make a table of the values of each of these expressions for different numerical values of the letter x.
We see that for all those values that were given to the letter x, the values of both expressions turned out to be equal. The same will be true for any other value of x.
To verify this, we transform the first expression. Based on the distribution law, we write:
Having performed the indicated operations on the numbers, we get:
So, the first expression, after its simplification, turned out to be exactly the same as the second expression.
Now it is clear that for any value of x, the values of both expressions are equal.
Expressions whose values are equal for any values of the letters included in them are called identically equal or identical.
Hence, they are identical expressions.
Let's make one important remark. Let's take expressions:
Having compiled a table similar to the previous one, we will make sure that both expressions, for any value of x, except for have equal numerical values. Only when the second expression is equal to 6, and the first loses its meaning, since the denominator is zero. (Recall that you cannot divide by zero.) Can we say that these expressions are identical?
We agreed earlier that each expression will be considered only for admissible values of letters, that is, for those values for which the expression does not lose its meaning. This means that here, when comparing two expressions, we take into account only those letter values that are valid for both expressions. Therefore, we must exclude the value. And since for all other values of x both expressions have the same numerical value, we have the right to consider them identical.
Based on what has been said, we give the following definition of identical expressions:
1. Expressions are called identical if they have the same numerical values for all admissible values of the letters included in them.
If we connect two identical expressions with an equal sign, then we get an identity. Means:
2. An identity is an equality that is true for all admissible values of the letters included in it.
We have already encountered identities before. So, for example, all equalities are identities, with which we expressed the basic laws of addition and multiplication.
For example, equalities expressing the commutative law of addition
and the associative law of multiplication
are valid for any values of letters. Hence, these equalities are identities.
All true arithmetic equalities are also considered identities, for example:
In algebra, one often has to replace an expression with another that is identical to it. Let, for example, it is required to find the value of the expression
We will greatly facilitate the calculations if we replace the given expression with an expression that is identical to it. Based on the distribution law, we can write:
But the numbers in brackets add up to 100. So, we have an identity:
Substituting 6.53 instead of a on the right side of it, we immediately (in the mind) find the numerical value (653) of this expression.
Replacing one expression with another, identical to it, is called the identical transformation of this expression.
Recall that any algebraic expression for any admissible values of letters is some
number. It follows from this that all the laws and properties of arithmetic operations that were given in the previous chapter are applicable to algebraic expressions. So, the application of the laws and properties of arithmetic operations transforms a given algebraic expression into an expression that is identical to it.
