Identical transformations. Converting expressions. Detailed theory (2020) Expression and their identical transformations

The arithmetic operation that is performed last when calculating the value of the expression is the "main" one.

That is, if you substitute some (any) numbers instead of letters, and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is factorized).

If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be canceled).

To fix the solution yourself, take a few examples:

Examples:

Solutions:

1. I hope you didn't rush to cut u right away? It was still not enough to "cut" units like this:

The first action should be factoring:

4. Addition and subtraction of fractions. Bringing fractions to a common denominator.

Addition and subtraction of ordinary fractions is a very familiar operation: we look for a common denominator, multiply each fraction by the missing factor and add / subtract the numerators.

Let's remember:

Answers:

1. Denominators and are mutually prime, that is, they have no common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:

2. Here the common denominator is:

3. Here, first of all, we turn mixed fractions into incorrect ones, and then - according to the usual scheme:

It is completely different if the fractions contain letters, for example:

Let's start simple:

a) Denominators do not contain letters

Here, everything is the same as with ordinary numeric fractions: we find the common denominator, multiply each fraction by the missing factor and add / subtract the numerators:

now in the numerator you can bring similar ones, if any, and decompose into factors:

Try it yourself:

Answers:

b) Denominators contain letters

Let's remember the principle of finding a common denominator without letters:

· First of all, we determine the common factors;

· Then write out all common factors once;

· And multiply them by all other factors that are not common.

To determine the common factors of the denominators, we first decompose them into prime factors:

Let's emphasize the common factors:

Now let's write out the common factors one time and add to them all non-common (not underlined) factors:

This is the common denominator.

Let's go back to the letters. The denominators are shown in exactly the same way:

· We decompose the denominators into factors;

· We determine common (identical) factors;

· Write out all common factors one time;

· We multiply them by all other factors, not common.

So, in order:

1) we decompose the denominators into factors:

2) we determine the common (identical) factors:

3) we write out all the common factors one time and multiply them by all the other (unstressed) factors:

So the common denominator is here. The first fraction must be multiplied by, the second by:

By the way, there is one trick:

For example: .

We see the same factors in the denominators, only all with different indicators. The common denominator will be:

to the extent

to the extent

to the extent

in degree.

Let's complicate the task:

How do you make fractions the same denominator?

Let's remember the basic property of a fraction:

Nowhere is it said that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because this is not true!

See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example,. What has been learned?

So, another unshakable rule:

When reducing fractions to a common denominator, use only multiplication!

But what must be multiplied by in order to receive?

Here on and multiply. And multiply by:

Expressions that cannot be decomposed into factors will be called "elementary factors".

For example, is an elementary factor. - also. But - no: it is factorized.

What do you think about expression? Is it elementary?

No, since it can be factorized:

(you already read about factorization in the topic "").

So, the elementary factors into which you expand the expression with letters are analogous to the prime factors into which you expand the numbers. And we will deal with them in the same way.

We see that both denominators have a factor. It will go to the common denominator in power (remember why?).

The factor is elementary, and it is not common to them, which means that the first fraction will simply have to be multiplied by it:

Another example:

Decision:

Before multiplying these denominators in a panic, you need to think about how to factor them into factors? They both represent:

Fine! Then:

Another example:

Decision:

As usual, factor the denominators. In the first denominator, we simply put it outside the brackets; in the second - the difference of squares:

It would seem that there are no common factors. But if you look closely, then they are so similar ... And the truth:

So let's write:

That is, it turned out like this: inside the parenthesis, we swapped the terms, and the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.

Now we bring to a common denominator:

Got it? Let's check now.

Tasks for an independent solution:

Answers:

Here we must remember one more - the difference between the cubes:

Please note that the denominator of the second fraction is not the "square of the sum" formula! The square of the sum would look like this:.

A is the so-called incomplete square of the sum: the second term in it is the product of the first and the last, and not their doubled product. The incomplete square of the sum is one of the factors in the expansion of the difference of cubes:

What if there are already three fractions?

The same thing! First of all, we will do so that the maximum number of factors in the denominators is the same:

Pay attention: if you change the signs inside one parenthesis, the sign in front of the fraction changes to the opposite. When we change the signs in the second parenthesis, the sign in front of the fraction is reversed again. As a result, it (the sign in front of the fraction) has not changed.

We write out the first denominator completely into the common denominator, and then add to it all the factors that have not yet been written, from the second, and then from the third (and so on, if there are more fractions). That is, it turns out like this:

Hmm ... With fractions, it's clear what to do. But what about the deuce?

It's simple: you know how to add fractions, right? It means that we need to make the deuce become a fraction! Remember: a fraction is a division operation (the numerator is divided by the denominator, in case you suddenly forgot). And nothing is easier than dividing a number by. In this case, the number itself will not change, but will turn into a fraction:

Exactly what is needed!

5. Multiplication and division of fractions.

Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:

Procedure

What is the procedure for calculating a numeric expression? Remember by counting the meaning of this expression:

Did you count it?

It should work out.

So, I remind you.

The first step is to calculate the degree.

The second is multiplication and division. If there are several multiplications and divisions at the same time, you can do them in any order.

And finally, we perform addition and subtraction. Again, in any order.

But: the expression in parentheses is evaluated out of order!

If several parentheses are multiplied or divided by each other, first calculate the expression in each of the parentheses, and then multiply or divide them.

What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. And when evaluating an expression, what is the first thing to do? That's right, calculate the parentheses. Well, we figured it out: first we calculate the inner brackets, then everything else.

So, the order of actions for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):

Okay, it's all simple.

But this is not the same as an expression with letters?

No, it's the same! Only, instead of arithmetic operations, you need to do algebraic ones, that is, the actions described in the previous section: bringing similar, addition of fractions, reduction of fractions, and so on. The only difference will be the effect of factoring polynomials (which we often use when working with fractions). Most often, for factoring, you need to use i or just put the common factor outside the brackets.

Usually our goal is to present an expression in the form of a work or a particular.

For example:

Let's simplify the expression.

1) First, we simplify the expression in brackets. There we have the difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:

It is impossible to simplify this expression anymore, all the factors here are elementary (do you still remember what this means?).

2) We get:

Multiplication of fractions: what could be easier.

3) Now you can shorten:

That's it. Nothing complicated, right?

Another example:

Simplify the expression.

First try to solve it yourself, and only then see the solution.

Decision:

First of all, let's define the order of actions.

First, we add the fractions in brackets, we get one instead of two fractions.

Then we will divide the fractions. Well, add the result with the last fraction.

I will enumerate the actions schematically:

Now I will show the whole process, coloring the current action in red:

1. If there are similar ones, they should be brought immediately. At whatever moment we have similar ones, it is advisable to bring them right away.

2. The same applies to the reduction of fractions: as soon as there is an opportunity to reduce, it must be used. The exception is fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.

Here are some tasks for you to solve on your own:

And promised at the very beginning:

Answers:

Solutions (concise):

If you have coped with at least the first three examples, then you have mastered the topic.

Now forward to learning!

TRANSFORMATION OF EXPRESSIONS. SUMMARY AND BASIC FORMULAS

Basic simplification operations:

• Bringing similar: to add (bring) such terms, you need to add their coefficients and assign the letter part.
• Factorization:factoring out the common factor, application, etc.
• Fraction reduction: the numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
  1) numerator and denominator factor out
  2) if there are common factors in the numerator and denominator, they can be crossed out.

  IMPORTANT: Only multipliers can be reduced!

• Addition and subtraction of fractions:
  ;
• Multiplication and division of fractions:
  ;

Identity conversions represent the work we do with numeric and literal expressions, as well as expressions that contain variables. We carry out all these transformations in order to bring the original expression to a form that will be convenient for solving the problem. We will consider the main types of identical transformations in this topic.

Identical conversion of an expression. What it is?

For the first time we meet with the concept of identical transformed we in algebra lessons in grade 7. At the same time, we first get acquainted with the concept of identically equal expressions. Let's understand the concepts and definitions to make the topic easier to understand.

Definition 1

Identical conversion of an expression - these are actions performed with the aim of replacing the original expression with an expression that will be identically equal to the original.

This definition is often used in an abbreviated form, in which the word "identical" is omitted. It is assumed that in any case we carry out the transformation of the expression in such a way as to obtain an expression identical to the original, and this does not need to be separately emphasized.

We will illustrate this definition examples.

Example 1

If we replace the expression x + 3 - 2 to an identical expression x + 1, then we will carry out the identical transformation of the expression x + 3 - 2.

Example 2

Replacing expression 2 a 6 with expression a 3 Is the identical transformation, while the replacement of the expression x on expression x 2 is not an identical transformation, since the expressions x and x 2 are not identically equal.

We draw your attention to the form of writing expressions when carrying out identical transformations. Typically, we write the original expression and the resulting expression as equality. So, writing x + 1 + 2 \u003d x + 3 means that the expression x + 1 + 2 has been reduced to the form x + 3.

The sequential execution of actions leads us to a chain of equalities, which is several identical transformations located in a row. So, we understand the notation x + 1 + 2 \u003d x + 3 \u003d 3 + x as the sequential implementation of two transformations: first, the expression x + 1 + 2 was brought to the form x + 3, and it - to the form 3 + x.

Identical transformations and ODU

A number of expressions that we begin to learn in grade 8 do not make sense for all values \u200b\u200bof the variables. Carrying out identical transformations in these cases requires us to pay attention to the range of permissible values \u200b\u200bof variables (ADV). Performing identical transformations can leave the ODZ unchanged or narrow it down.

Example 3

When jumping from expression a + (- b) to the expression a - b variable range a and b remains the same.

Example 4

Go from expression x to expression x 2 x leads to a narrowing of the range of admissible values \u200b\u200bof the variable x from the set of all real numbers to the set of all real numbers, from which zero was excluded.

Example 5

Identical conversion of an expression x 2 xexpression x leads to the expansion of the range of admissible values \u200b\u200bof the variable x from the set of all real numbers except zero to the set of all real numbers.

Narrowing or expanding the range of permissible values \u200b\u200bof variables when carrying out identical transformations is important in solving problems, since it can affect the accuracy of calculations and lead to errors.

Basic identity transformations

Let's now see what identical transformations are and how they are performed. Let us single out those types of identical transformations, with which we have to deal most often, into the main group.

In addition to the basic identical transformations, there are a number of transformations that relate to expressions of a specific type. For fractions, these are methods of reduction and reduction to a new denominator. For expressions with roots and powers, all actions that are performed based on the properties of roots and powers. For logarithmic expressions, actions that are performed based on the properties of logarithms. For trigonometric expressions all actions using trigonometric formulas... All these private transformations are detailed in separate topics that can be found on our resource. In this regard, we will not dwell on them in this article.

Let's move on to considering the basic identical transformations.

Permutation of terms, factors

Let's start by rearranging the terms. We deal with this identical transformation most often. And the following statement can be considered the basic rule here: in any sum, the permutation of the terms in places does not affect the result.

This rule is based on the displacement and combination properties of addition. These properties allow us to rearrange the terms in places and obtain expressions that are identically equal to the original ones. That is why the permutation of the terms in places in the sum is the identity transformation.

Example 6

We have the sum of three terms 3 + 5 + 7. If we swap the terms 3 and 5, then the expression takes the form 5 + 3 + 7. There are several options for rearranging the terms in this case. All of them lead to obtaining expressions that are identical to the original one.

Not only numbers, but also expressions can act as terms in the sum. Just like the numbers, they can be rearranged without affecting the final result of the calculations.

Example 7

In the sum of three terms 1 a + b, a 2 + 2 a + 5 + a 7 a 3 and - 12 a of the form 1 a + b + a 2 + 2 a + 5 + a 7 a 3 + ( - 12) · a terms can be rearranged, for example, as follows (- 12) · a + 1 a + b + a 2 + 2 · a + 5 + a 7 · a 3. In turn, you can rearrange the terms in the denominator of the fraction 1 a + b, and the fraction will take the form 1 b + a. And the expression under the root sign a 2 + 2 a + 5 is also the sum in which the terms can be swapped.

In the same way as the terms, in the original expressions, you can swap the factors and obtain identically correct equations. This action is governed by the following rule:

Definition 2

In a product, rearranging the multipliers in places does not affect the calculation result.

This rule is based on the displacement and combination properties of multiplication, which confirm the correctness of the identical transformation.

Example 8

Composition 3 5 7 permutation of factors can be represented in one of the following forms: 5 3 7, 5 7 3, 7 3 5, 7 5 3 or 3 7 5.

Example 9

Rearranging the factors in the product x + 1 x 2 - x + 1 x gives x 2 - x + 1 x x + 1

Expanding brackets

The parentheses can contain numeric and variable expressions. These expressions can be converted into identically equal expressions, in which there will be no parentheses at all or there will be fewer of them than in the original expressions. This way of converting expressions is called parenthesis expansion.

Example 10

Let's perform actions with brackets in an expression of the form 3 + x - 1 x in order to get an identically correct expression 3 + x - 1 x.

The expression 3 x - 1 + - 1 + x 1 - x can be converted to the identically equal expression without parentheses 3 x - 3 - 1 + x 1 - x.

We have detailed the rules for converting expressions with brackets in the topic "Expanding brackets", which is posted on our resource.

Grouping of terms, factors

In cases when we are dealing with three or more terms, we can resort to such a form of identical transformations as the grouping of terms. This method of transformations means combining several terms into a group by rearranging them and enclosing them in parentheses.

During the grouping, the terms are interchanged so that the terms to be grouped appear side by side in the expression. They can then be enclosed in parentheses.

Example 11

Let's take the expression 5 + 7 + 1 ... If we group the first term with the third, we get (5 + 1) + 7 .

The grouping of factors is carried out similarly to the grouping of terms.

Example 12

In the work 2 3 4 5 we can group the first factor with the third, and the second with the fourth, and we arrive at the expression (2 4) (3 5)... And if we grouped the first, second and fourth factors, we would get the expression (2 3 5) 4.

The terms and factors that are grouped can be represented by both prime numbers and expressions. The grouping rules were discussed in detail in the topic "Grouping of terms and factors".

Replacing differences with sums, partial products and vice versa

Replacing the differences with sums became possible thanks to our acquaintance with opposite numbers. Now subtracting from a number a numbers b can be viewed as an addition to the number a numbers - b... Equality a - b \u003d a + (- b)can be considered fair and on its basis to replace the differences with sums.

Example 13

Let's take the expression 4 + 3 − 2 , in which the difference of numbers 3 − 2 we can write as the sum 3 + (− 2) ... We get 4 + 3 + (− 2) .

Example 14

All differences in expression 5 + 2 x - x 2 - 3 x 3 - 0, 2 can be replaced by sums like 5 + 2 x + (- x 2) + (- 3 x 3) + (- 0, 2).

We can go over to the sums from any differences. Similarly, we can make the reverse replacement.

Replacing division with multiplication by the reciprocal of the divisor is made possible by the concept of reciprocal numbers. This transformation can be written by the equality a: b \u003d a (b - 1).

This rule was the basis for the rule for dividing ordinary fractions.

Example 15

Private 1 2: 3 5 can be replaced by a product of the form 1 2 5 3.

Likewise, by analogy, division can be replaced by multiplication.

Example 16

In the case of the expression 1 + 5: x: (x + 3)replace division with x can be multiplied by 1 x... Division by x + 3 we can replace by multiplication by 1 x + 3... The transformation allows us to get an expression identical to the original one: 1 + 5 · 1 x · 1 x + 3.

The replacement of multiplication by division is carried out according to the scheme a b \u003d a: (b - 1).

Example 17

In the expression 5 x x 2 + 1 - 3, multiplication can be replaced by division as 5: x 2 + 1 x - 3.

Performing actions on numbers

Performing actions with numbers obeys the rule of order of actions. First, actions are performed with powers of numbers and roots of numbers. After that, we replace logarithms, trigonometric and other functions with their values. Then the actions in parentheses are performed. And then all other actions can be carried out from left to right. It is important to remember that multiplication and division are performed before addition and subtraction.

Operations with numbers allow you to convert the original expression to the identical equal to it.

Example 18

Rewrite the expression 3 · 2 3 - 1 · a + 4 · x 2 + 5 · x, performing all possible actions with the numbers.

Decision

First of all, let's pay attention to the degree 2 3 and root 4 and calculate their values: 2 3 = 8 and 4 \u003d 2 2 \u003d 2.

Substitute the obtained values \u200b\u200binto the original expression and get: 3 · (8 - 1) · a + 2 · (x 2 + 5 · x).

Now let's perform the actions in brackets: 8 − 1 = 7 ... And move on to the expression 3 7 a + 2 (x 2 + 5 x).

It remains for us to perform the multiplication of numbers 3 and 7 ... We get: 21 a + 2 (x 2 + 5 x).

Answer: 3 2 3 - 1 a + 4 x 2 + 5 x \u003d 21 a + 2 (x 2 + 5 x)

Actions on numbers can be preceded by other kinds of identical transformations, such as grouping numbers or expanding parentheses.

Example 19

Let's take the expression 3 + 2 (6: 3) x (y 3 4) - 2 + 11.

Decision

The first step is to replace the quotient in brackets 6: 3 on its value 2 ... We get: 3 + 2 2 x (y 3 4) - 2 + 11.

Let's expand the brackets: 3 + 2 2 x (y 3 4) - 2 + 11 \u003d 3 + 2 2 x y 3 4 - 2 + 11.

Let's group the numerical factors in the product, as well as the terms that are numbers: (3 - 2 + 11) + (2 2 4) x y 3.

Let's perform the actions in brackets: (3 - 2 + 11) + (2 2 4) x y 3 \u003d 12 + 16 x y 3

Answer: 3 + 2 (6: 3) x (y 3 4) - 2 + 11 \u003d 12 + 16 x y 3

If we work with numerical expressions, then the goal of our work will be to find the meaning of the expression. If we transform expressions with variables, then the goal of our actions will be to simplify the expression.

Factor out the common factor

In cases where the terms in the expression have the same factor, then we can take this common factor outside the brackets. To do this, we first need to represent the original expression as the product of the common factor and the expression in brackets, which consists of the original terms without the common factor.

Example 20

Numerically 2 7 + 2 3 we can take out the common factor 2 brackets and get an identically correct expression of the form 2 (7 + 3).

You can refresh your memory of the rules for putting the common factor outside the brackets in the corresponding section of our resource. The material discusses in detail the rules for putting the common factor outside the brackets and provides numerous examples.

Reduction of similar terms

Now let's move on to the sums that contain similar terms. There are two possible options: the sums containing the same terms, and the sums, the terms of which differ in numerical coefficient. Actions with sums containing such terms are called the reduction of such terms. It is carried out as follows: we take out the general letter part outside the brackets and calculate the sum of the numerical coefficients in brackets.

Example 21

Consider the expression 1 + 4 x - 2 x... We can put the literal part x outside the brackets and get the expression 1 + x (4 - 2)... Let's calculate the value of the expression in parentheses and get the sum of the form 1 + x · 2.

Replacing numbers and expressions with identically equal expressions

The numbers and expressions from which the original expression is composed can be replaced with identically equal expressions. Such a transformation of the original expression leads to an expression identically equal to it.

Example 22 Example 23

Consider the expression 1 + a 5, in which we can replace the degree of a 5 with an identically equal product, for example, of the form a a 4... This will give us the expression 1 + a a 4.

The transformation performed is artificial. It only makes sense in preparation for other transformations.

Example 24

Consider the transformation of the sum 4 x 3 + 2 x 2... Here the term 4 x 3 we can represent as a work 2 x 2 2 x... As a result, the original expression takes the form 2 x 2 2 x + 2 x 2... Now we can select the common factor 2 x 2 and put it outside the brackets: 2 x 2 (2 x + 1).

Add and subtract the same number

Adding and subtracting the same number or expression at the same time is an artificial technique for transforming expressions.

Example 25

Consider the expression x 2 + 2 x... We can add or subtract one from it, which will allow us to carry out another identical transformation in the future - to select the square of the binomial: x 2 + 2 x \u003d x 2 + 2 x + 1 - 1 \u003d (x + 1) 2 - 1.

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“Identities. Identical transformation of expressions ”.

Lesson objectives

Educational:

to acquaint and firstly consolidate the concepts of "identically equal expressions", "identity", "identical transformations";

consider ways of proving identities, contribute to the development of skills for proving identities;

to check the assimilation of the passed material by the students, to form the skills of using the learned for the perception of the new.

Developing : develop thinking, speech of students.

Educational : to educate diligence, accuracy, correctness of recording the solution of exercises.

Lesson type: learning new material

Equipment : Multimedia board, whiteboard, textbook, workbook.

P lahn lesson

Organizational moment (aim students at the lesson)

Homework check (error correction)

Oral exercises

Studying new material (Acquaintance and primary consolidation of the concepts of "identity", "identical transformations").

Training exercises (Formation of the concepts of "identity", "identical transformations").

Summing up the lesson (Summarize the theoretical information obtained in the lesson).

Homework message (Explain homework content)

During the classes

I. Organizational moment.

Homework check.

Homework questions.

Analysis of the solution at the blackboard.

Mathematics is needed
You can't live without her
We teach, we teach, friends,
What do we remember from the morning?

II ... Oral exercises.

Let's do a warm-up.

Addition result. (Amount)

How many numbers do you know? (Ten)

One hundredth of the number. (Percent)

Division result? (Private)

The smallest natural number? (1)

Is it possible when dividing natural numbers get zero? (not)

What is the sum of the numbers from -200 to 200? (0)

What is the largest negative integer. (-1)

What number cannot be divided by? (0)

The result of the multiplication? (Composition)

Largest two-digit number? (99)

What is the product from -200 to 200? (0)

Subtraction result. (Difference)

How many grams are in a kilogram? (1000)

The displacement property of addition. (The sum does not change from the rearrangement of the places of the terms)

The travel property of multiplication. (The product does not change from permutation of the multipliers)

Combination property of addition. (To add a number to the sum of two numbers, you can add the sum of the second and third to the first number)

Combination property of multiplication. (to multiply the product of two numbers by the third number, you can multiply the first number by the product of the second and third)

Distribution property. (To multiply a number by the sum of two numbers, you can multiply that number by each term and add the results)

III ... Learning new material .

Teacher. Find the value of the expressions for x \u003d 5 and y \u003d 4

3 (x + y) \u003d 3 (5 + 4) \u003d 3 * 9 \u003d 27

3x + 3y \u003d 3 * 5 + 3 * 4 \u003d 27

We got the same result. From the distribution property it follows that, in general, for any values \u200b\u200bof the variables, the values \u200b\u200bof the expressions 3 (x + y) and 3x + 3y are equal.

Consider now the expressions 2x + y and 2xy. For x \u003d 1 and y \u003d 2, they take equal values:

2x + y \u003d 2 * 1 + 2 \u003d 4

2xy \u003d 2 * 1 * 2 \u003d 4

However, you can specify values \u200b\u200bfor x and y such that the values \u200b\u200bof these expressions are not equal. For example, if x \u003d 3, y \u003d 4, then

2x + y \u003d 2 * 3 + 4 \u003d 10

2xy \u003d 2 * 3 * 4 \u003d 24

Definition: Two expressions, the values \u200b\u200bof which are equal for any values \u200b\u200bof the variables, are called identically equal.

The expressions 3 (x + y) and 3x + 3y are identically equal, but the expressions 2x + y and 2xy are not identically equal.

The equality 3 (x + y) and 3x + 3y is true for any values \u200b\u200bof x and y. Such equalities are called identities.

Definition: Equality, true for any values \u200b\u200bof the variables, is called identity.

True numerical equalities are also considered identities. We have already met with identities. Identities are equalities that express the basic properties of actions on numbers (Students comment on each property, pronouncing it).

a + b \u003d b + a ab \u003d ba (a + b) + c \u003d a + (b + c) (ab) c \u003d a (bc) a (b + c) \u003d ab + ac

Other examples of identities (Students comment on each property by speaking.)

a + 0 \u003d a

a * 1 \u003d a

a + (-a) \u003d 0

and * (- b ) = - ab

a - b = a + (- b )

(- a ) * (- b ) = ab

Definition: Replacing one expression with another, identically equal expression, is called an identity transformation or simply an expression transformation.

Teacher:

Identical transformations of expressions with variables are performed based on the properties of actions on numbers.

Identical transformations of expressions are widely used in calculating the values \u200b\u200bof expressions and solving other problems. You have already performed some identical transformations, for example, casting similar terms, expanding parentheses. Let us recall the rules for these transformations:

Students:

To bring such terms, you need to add their coefficients and multiply the result by the total letter part;

If there is a plus sign in front of the brackets, then the brackets can be omitted, keeping the sign of each term enclosed in brackets;

If there is a minus sign in front of the brackets, then the brackets can be omitted by changing the sign of each term enclosed in brackets.

Teacher:

Example 1. Let us present similar terms

5x + 2x-3x \u003d x (5 + 2-3) \u003d 4x

Which rule did we use?

Pupil:

We have used the rule for the reduction of such terms. This transformation is based on the distribution property of multiplication.

Teacher:

Example 2. Let's expand the brackets in the expression 2a + (b-3 c) = 2 a + b – 3 c

We applied the rule for expanding brackets preceded by a plus sign.

Pupil:

The performed transformation is based on the combinational property of addition.

Teacher:

Example 3. Let's expand the brackets in the expression a - (4b - s) \u003da – 4 b + c

We used the rule of opening brackets, preceded by a minus sign.

What property is this transformation based on?

Pupil:

The transformation performed is based on the distribution property of multiplication and the combination property of addition.

IV ... Training exercises

(Before starting, we spend a physical education

We got up quickly and smiled.

They stretched higher and higher.

Well, straighten your shoulders,

Raise, lower.

Turn right, left,

They sat down, got up. They sat down, got up.

And they ran on the spot.

(Well done, have a seat).

Let's carry out mini independent work - correspondences, And those who believe that the topic is well mastered - decide online testing.

1) 5 (3x -2) - (4x + 9) A) 5-10: x

2) 5x-4 (2x-5) +5 B) 11x -19

3) (5x-10): x B) 3x + 25

4) 11x-4 (x - 3) + 5x D) -3x + 25

D) 12x +12

V ... Lesson summary .

The teacher asks questions, and the students answer them as they wish.

What two expressions are called identically equal? Give examples.

What equality is called identity? Give an example.

What identical transformations do you know?

VI ... Homework ... item 5, find old identical expressions using the Internet

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Slide captions:

Identities. Identical transformations of expressions. 7th grade.

Find the value of the expressions at x \u003d 5 and y \u003d 4 3 (x + y) \u003d 3 (5 + 4) \u003d 3 * 9 \u003d 27 3x + 3y \u003d 3 * 5 + 3 * 4 \u003d 27 Find the value of the expressions at x \u003d 6 and y \u003d 5 3 (x + y) \u003d 3 (6 + 5) \u003d 3 * 11 \u003d 33 3x + 3y \u003d 3 * 6 + 3 * 5 \u003d 33

CONCLUSION: We got the same result. From the distribution property it follows that, in general, for any values \u200b\u200bof the variables, the values \u200b\u200bof the expressions 3 (x + y) and 3x + 3y are equal. 3 (x + y) \u003d 3x + 3y

Consider now the expressions 2x + y and 2xy. for x \u003d 1 and y \u003d 2 they take equal values: 2x + y \u003d 2 * 1 + 2 \u003d 4 2xy \u003d 2 * 1 * 2 \u003d 4 for x \u003d 3, y \u003d 4 the values \u200b\u200bof the expressions are different 2x + y \u003d 2 * 3 + 4 \u003d 10 2xy \u003d 2 * 3 * 4 \u003d 24

CONCLUSION: Expressions 3 (x + y) and 3x + 3y are identically equal, and expressions 2x + y and 2xy are not identically equal. Definition: Two expressions whose values \u200b\u200bare equal for any values \u200b\u200bof the variables are called identically equal.

Identity The equality 3 (x + y) and 3x + 3y is true for any values \u200b\u200bof x and y. Such equalities are called identities. Definition: Equality, true for any values \u200b\u200bof the variables, is called identity. True numerical equalities are also considered identities. We have already met with identities.

Identities are equalities that express the basic properties of actions on numbers. a + b \u003d b + a ab \u003d ba (a + b) + c \u003d a + (b + c) (ab) c \u003d a (bc) a (b + c) \u003d ab + ac

Other examples of identities can be given: a + 0 \u003d a a * 1 \u003d a a + (-a) \u003d 0 a * (- b) \u003d - ab a- b \u003d a + (- b) (-a) * ( -b) \u003d ab Replacing one expression with another, identically equal to it, is called an identity conversion or simply an expression conversion.

To bring such terms, you need to add their coefficients and multiply the result by the total letter part. Example 1. Let's give similar terms 5x + 2x-3x \u003d x (5 + 2-3) \u003d 4x

If there is a plus sign in front of the brackets, the brackets can be omitted, keeping the sign of each term enclosed in brackets. Example 2. Expand the brackets in the expression 2a + (b -3 c) \u003d 2 a + b - 3 c

If there is a minus sign in front of 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 subject: methodological developments, presentations and notes

Methodology for preparing students for the exam in the section "Expressions and expression transformation"

This project was developed with the aim of preparing students for the state exams in the 9th grade and later for the unified state exam in the 11th grade ....

In the course of studying algebra, we came across the concepts of a polynomial (for example ($ yx $, $ \\ 2x ^ 2-2x $, etc.) and an algebraic fraction (for example $ \\ frac (x + 5) (x) $, $ \\ frac (2x ^ 2) (2x ^ 2-2x) $, $ \\ \\ frac (xy) (yx) $, etc.) The similarity of these concepts is that both in polynomials and in algebraic fractions there are variables and numerical values, arithmetic actions: addition, subtraction, multiplication, raising to a power.The difference between these concepts is that in polynomials there is no division by a variable, but in algebraic fractions, division by a variable can be done.

Both polynomials and algebraic fractions in mathematics are called rational algebraic expressions. But polynomials are entire rational expressions, and algebraic fractions are fractionally rational expressions.

You can get an entire algebraic expression from a fractional-rational expression using the identical transformation, which in this case will be the main property of the fraction - the reduction of fractions. Let's check it out in practice:

Example 1

Perform transformation: $ \\ \\ frac (x ^ 2-4x + 4) (x-2) $

Decision: This fractional-rational equation can be transformed by using the basic property of the fractional-reduction, i.e. dividing the numerator and denominator by the same number or expression other than $ 0 $.

This fraction cannot be canceled right away, it is necessary to transform the numerator.

We transform the expression in the numerator of the fraction, for this we use the formula for the square of the difference: $ a ^ 2-2ab + b ^ 2 \u003d ((a-b)) ^ 2 $

The fraction looks like

\\ [\\ frac (x ^ 2-4x + 4) (x-2) \u003d \\ frac (x ^ 2-4x + 4) (x-2) \u003d \\ frac (((x-2)) ^ 2) ( x-2) \u003d \\ frac (\\ left (x-2 \\ right) (x-2)) (x-2) \\]

Now we see that both the numerator and the denominator have a common factor - this is the expression $ x-2 $, by which we will cancel the fraction

\\ [\\ frac (x ^ 2-4x + 4) (x-2) \u003d \\ frac (x ^ 2-4x + 4) (x-2) \u003d \\ frac (((x-2)) ^ 2) ( x-2) \u003d \\ frac (\\ left (x-2 \\ right) (x-2)) (x-2) \u003d x-2 \\]

After reduction, we got that the original fractional-rational expression $ \\ frac (x ^ 2-4x + 4) (x-2) $ became a polynomial $ x-2 $, i.e. whole rational.

Now let's pay attention to the fact that the expressions $ \\ frac (x ^ 2-4x + 4) (x-2) $ and $ x-2 \\ $ can be considered identical not for all values \u200b\u200bof the variable, since in order for the fractional rational expression to exist and it was possible to reduce by the polynomial $ x-2 $, the denominator of the fraction must not be equal to $ 0 $ (as well as the factor by which we reduce. In this example, the denominator and the factor coincide, but this is not always the case).

The values \u200b\u200bof the variable at which the algebraic fraction will exist are called the admissible values \u200b\u200bof the variable.

Let us put a condition on the denominator of the fraction: $ x-2 ≠ 0 $, then $ x ≠ 2 $.

Hence, the expressions $ \\ frac (x ^ 2-4x + 4) (x-2) $ and $ x-2 $ are identical for all values \u200b\u200bof the variable, except for $ 2 $.

Definition 1

Identically equal expressions are those that are equal for all allowable values \u200b\u200bof the variable.

Identical transformation is any replacement of the original expression with an identically equal to it. Such transformations include performing actions: addition, subtraction, multiplication, taking a common factor out of a bracket, reducing algebraic fractions to a common denominator, reducing algebraic fractions, reducing similar terms, etc. It should be borne in mind that a number of transformations, such as reduction, reduction of similar terms can change the permissible values \u200b\u200bof the variable.

Techniques used to prove identities

Bring the left side of the identity to the right or vice versa using the identity transformations

Reduce both sides to the same expression using identical transformations

Move expressions in one part of the expression to another and prove that the resulting difference is $ 0 $

Which of the above methods to use to prove a given identity depends on the original identity.

Example 2

Prove the identity $ ((a + b + c)) ^ 2- 2 (ab + ac + bc) \u003d a ^ 2 + b ^ 2 + c ^ 2 $

Decision: To prove this identity, we use the first of the above methods, namely, we transform the left side of the identity to its equality with the right.

Consider the left side of the identity: $ \\ ((a + b + c)) ^ 2- 2 (ab + ac + bc) $ - it is the difference of two polynomials. The first polynomial is the square of the sum of three terms. To square the sum of several terms, we use the formula:

\\ [((a + b + c)) ^ 2 \u003d a ^ 2 + b ^ 2 + c ^ 2 + 2ab + 2ac + 2bc \\]

To do this, we need to multiply the number by a polynomial. Recall that for this we need to multiply the common factor outside the brackets by each term of the polynomial in the brackets. Then we get:

$ 2 (ab + ac + bc) \u003d 2ab + 2ac + 2bc $

Now back to the original polynomial, it will take the form:

$ ((a + b + c)) ^ 2- 2 (ab + ac + bc) \u003d \\ a ^ 2 + b ^ 2 + c ^ 2 + 2ab + 2ac + 2bc- (2ab + 2ac + 2bc) $

Note that before the parenthesis there is a "-" sign, which means that when the parentheses are opened, all the characters that were in the parentheses are reversed.

$ ((a + b + c)) ^ 2- 2 (ab + ac + bc) \u003d \\ a ^ 2 + b ^ 2 + c ^ 2 + 2ab + 2ac + 2bc- (2ab + 2ac + 2bc) \u003d a ^ 2 + b ^ 2 + c ^ 2 + 2ab + 2ac + 2bc-2ab-2ac-2bc $

Given similar terms, we get that the monomials $ 2ab $, $ 2ac $, $ \\ 2bc $ and $ -2ab $, $ - 2ac $, $ -2bc $ are mutually canceled, i.e. their sum is $ 0.

$ ((a + b + c)) ^ 2- 2 (ab + ac + bc) \u003d \\ a ^ 2 + b ^ 2 + c ^ 2 + 2ab + 2ac + 2bc- (2ab + 2ac + 2bc) \u003d a ^ 2 + b ^ 2 + c ^ 2 + 2ab + 2ac + 2bc-2ab-2ac-2bc \u003d a ^ 2 + b ^ 2 + c ^ 2 $

So, by means of identical transformations, we got the identical expression on the left side of the original identity

$ ((a + b + c)) ^ 2- 2 (ab + ac + bc) \u003d \\ a ^ 2 + b ^ 2 + c ^ 2 $

Note that the resulting expression shows that the original identity is true.

Note that in the original identity, all values \u200b\u200bof the variable are admissible, which means that we have proved the identity using identical transformations, and it is true for all admissible values \u200b\u200bof the variable.