Identical transformations of expressions. Identically equal expressions: definition, examples

The main properties of addition and multiplication of numbers.

Moving property of addition: the amount of the sum is not changed from the permutation of the terms. For any numbers a and b is true equality

The combination property of addition: to add the third number to the sum of two numbers, you can add the second and third amount to the first number. For any numbers a, b and c are true equality

Multiplication Property: From the rearrangement of multipliers, the product value does not change. For any numbers A, B and C are true equality

The combination property of multiplication: To multiply the work of two numbers to multiply by the third number, you can multiply the first number to the work of the second and third.

For any numbers A, B and C are true equality

Distribution Property: To multiply a number of amount, you can multiply this number to each aligned and folded the results. For any numbers a, b and c are true equality

From the recessive and combination properties of the addition, it follows: in any amount you can somehow rearrange the components and randomly combine them into groups.

Example 1 Calculate the amount of 1.23 + 13.5 + 4.27.

To do this, it is convenient to merge the first term with the third. We get:

1,23+13,5+4,27=(1,23+4,27)+13,5=5,5+13,5=19.

From the Movement and Fashion Properties of Multiplication, it is necessary: \u200b\u200bin any product, you can quickly rearrange the multipliers and randomly combine them into groups.

Example 2 Find the value of the product 1.8 · 0.25 · 64 · 0.5.

By combining the first factor with the fourth, and the second on the third, we will have:

1.8 · 0.25 · 64 · 0.5 \u003d (1.8 · 0.5) · (0.25 · 64) \u003d 0.9 · 16 \u003d 14.4.

The distribution property is valid and in the case when the number is multiplied by the amount of three and more components.

For example, for any numbers a, b, c and d, equality is true

a (B + C + D) \u003d AB + AC + AD.

We know that subtraction can be replaced by adding, adding to a reduced number opposite to subtractable:

This allows a numeric expression of the type A-B to consider the sum of the numbers a and -b, the numerical expression of the form A + B-C-D is considered the sum of the number a, b, -c, -d, etc. The considered properties of actions are fair and for such sums.

Example 3 Find the expression value of 3.27-6.5-2.5 + 1.73.

This expression is the sum of numbers 3.27, -6.5, -2.5 and 1.73. Applying the addition properties, we obtain: 3,27-6.5-2.5 + 1.73 \u003d (3.27 + 1.73) + (- 6.5-2.5) \u003d 5 + (- 9) \u003d -four.

Example 4 Calculate the work 36 · ().

The multiplier can be considered as the sum of numbers and -. Using the distribution property of multiplication, we get:

36 () \u003d 36 · -36 · \u003d 9-10 \u003d -1.

Identities

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

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

Find the values \u200b\u200bof expressions 3 (x + y) and 3x + 3y at x \u003d 5, y \u003d 4:

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

3X + 3Y \u003d 3 · 5 + 3 · 4 \u003d 15 + 12 \u003d 27.

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

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

However, you can specify such values \u200b\u200bX and Y, in which the values \u200b\u200bof these expressions are not equal. For example, if x \u003d 3, y \u003d 4, then

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

Equality 3 (x + y) \u003d x + 3y, faithful for any x and y values, is a identity.

The identities of the faithful numerical equality are also considered.

So, the identities are equalities expressing the main properties of actions above the numbers:

a + b \u003d b + a, (a + b) + c \u003d a + (b + c),

aB \u003d BA, (AB) C \u003d A (BC), A (B + C) \u003d AB + AC.

Other examples of identities can also be given:

a + 0 \u003d a, a + (- a) \u003d 0, a - b \u003d a + (- b),

a · 1 \u003d a, a · (-b) \u003d - AB, (-a) (- b) \u003d AB.

Identical transformations of expressions

The replacement of one expression to another, identically equal to it is called the identical conversion or simply by the transformation of the expression.

The identity transformations of expressions with variables are based on the properties of the number of numbers.

To find the value of the XY-XZ expression at the specified x, y, z values, three actions must be performed. For example, at x \u003d 2.3, y \u003d 0.8, z \u003d 0,2 we get:

xy - xz \u003d 2.3 · 0.8-2.3 · 0.2 \u003d 1.84-0.46 \u003d 1.38.

This result can be obtained by performing only two actions, if you use the expression X (Y-Z), identically equal to the expression XY-XZ:

xy - xz \u003d 2.3 (0.8-0.2) \u003d 2.3 · 0.6 \u003d 1.38.

We simplified the calculation, replacing the expression XY-XZ identically equal to the expression X (Y-Z).

The identity transformations of expressions are widely used when calculating the values \u200b\u200bof expressions and solving other tasks. Some identical transformations have already had to be performed, for example, bringing such terms, disclosing brackets. Recall the rules for performing these transformations:

to bring similar terms, it is necessary to fold their coefficients and the result is multiplied by a common letterway;

if the "Plus" sign is standing in front of the brackets, then the brackets can be omitted, while retaining the sign of each term enclosed in the bracket;

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 the bracket.

Example 1 We present similar terms in the amount of 5x + 2x-3x.

We use the rule of bringing similar terms:

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

This conversion is based on the distribution property of multiplication.

Example 2 Recall brackets in the expression 2a + (B-3C).

Applying the disclosure rule of the brackets, in front of which the "Plus" sign is:

2a + (B-3C) \u003d 2A + B-3C.

The transformation is based on a combination property of addition.

Example 3 Recall brackets in the expression A- (4B-C).

We use the rules for disclosing brackets, facing the minus sign:

a- (4B-C) \u003d A-4B + C.

The conversion performed is based on the distribution property of multiplication and the combination property of addition. Show it. Imagine in this expression the second term - (4B-C) as a work (-1) (4B-C):

a- (4B-C) \u003d A + (- 1) (4B-C).

Applying the specified properties of actions, we get:

a- (4B-C) \u003d a + (- 1) (4B-C) \u003d A + (- 4B + C) \u003d A-4B + C.

The numbers and expressions of which are composed of the original expression, can be replaced by identically equal expressions. Such transformation of the initial expression leads to an identically equal expression.

For example, in the expression 3 + x, the number 3 can be replaced with the amount of 1 + 2, and the expression (1 + 2) + x, which is identically equal to the initial expression. Another example: in the expression 1 + A 5 degree A 5 can be replaced by an identically equal to it, for example, the form A · A 4. This will give us an expression 1 + A · A 4.

This transformation is undoubtedly artificially, and is usually prepared for any further transformations. For example, in the amount 4 · x 3 + 2 · x 2, given the properties of the degree, the term 4 · x 3 can be represented as a piece of 2 · x 2 · 2 · x. After such a transformation, the initial expression will take a form 2 · x 2 · 2 · x + 2 · x 2. Obviously, the components of the total multiplier 2 · x 2 have the components in the resulting amount, so we can perform the following transformation - exercise for brackets. After it, we will come to expression: 2 · x 2 · (2 \u200b\u200b· x + 1).

Adjustment and subtraction of the same number

Another artificial expression transformation is the addition and simultaneous subtraction of the same number or expression. Such a transformation is identical, as it is, in fact, equivalent to the increase of zero, and the reduction of zero does not change the values.

Consider an example. Take the expression x 2 + 2 · x. If you add a unit to it and take a unit, it will further complete another identity conversion - select the square of bounce: x 2 + 2 · x \u003d x 2 + 2 · x + 1-1 \u003d (x + 1) 2 -1.

Bibliography.

Algebra: studies. for 7 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov]; Ed. S. A. Telikovsky. - 17th ed. - M.: Enlightenment, 2008. - 240 s. : IL. - ISBN 978-5-09-019315-3.
Algebra: studies. For 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov]; Ed. S. A. Telikovsky. - 16th ed. - M.: Enlightenment, 2008. - 271 p. : IL. - ISBN 978-5-09-019243-9.
Mordkovich A. G. Algebra. 7th grade. In 2 tsp. 1. Tutorial for students of general educational institutions / A. Mordkovich. - 17th ed., Extras - M.: Mnemozina, 2013. - 175 p.: Il. ISBN 978-5-346-02432-3.

Consider two equalities:

1. A 12 * A 3 \u003d A 7 * A 8

This equality will be performed at any values \u200b\u200bof the variable a. The area of \u200b\u200bpermissible values \u200b\u200bfor that equality will be all many real numbers.

2. A 12: A 3 \u003d A 2 * A 7.

This inequality will be performed for all values \u200b\u200bof the variable A, except for zero. The area of \u200b\u200bpermissible values \u200b\u200bfor this inequality will be all many real numbers, except for zero.

Each of these equalities can be argued that it will be true for any permissible values \u200b\u200bof variables a. Such equalities in mathematics are called identities.

The concept of identity

The identity is equality faithful in any permissible values \u200b\u200bof variables. If in this equality to substitute, instead of variables, any valid values, then the correct numerical equality should be obtained.

It is worth noting that faithful numeric equality are also identities. Identities, for example, will be the properties of actions on numbers.

3. A + B \u003d B + A;

4. A + (B + C) \u003d (A + B) + C;

6. A * (B * C) \u003d (A * B) * C;

7. A * (B + C) \u003d A * B + A * C;

11. A * (- 1) \u003d -a.

If two expressions with any valid variables are respectively equal, then such expressions are called identically equal. Below are several examples of identically equal expressions:

1. (A 2) 4 and a 8;

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

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

We can always replace one expression by any other expression, identically equal to the first. Such a replacement will be identical conversion.

Examples of identities

Example 1: The following equalities are identities:

1. A + 5 \u003d 5 + A;

2. A * (- b) \u003d -a * b;

3. 3 * A * 3 * B \u003d 9 * A * B;

Not all the above expressions will be identities. Of these equalities, the identities are only 1.2 and 3 equality. Whatever numbers we do not put in them, instead of variables a and b, we still have faithful numeric equality.

But 4 equality is no longer identity. Because not with all permissible values, this equality will be performed. For example, at values \u200b\u200bA \u003d 5 and B \u003d 2, the following result will be:

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

§ 2. Identical expressions, identity. Identical transformation of expression. Proof of identity

Find the values \u200b\u200bof expressions 2 (x - 1) 2x - 2 for these values \u200b\u200bof the variable x. Results We write to the table:

It can be concluded that the values \u200b\u200bof expressions 2 (x - 1) 2x - 2 for each given value of the variable x are equal to each other. According to the distributional property of multiplication relative to subtraction 2 (x - 1) \u003d 2x - 2. Therefore, for any other value of the variable x, the expression value 2 (x - 1) 2x - 2 will also be equal to each other. Such expressions are called identically equal.

For example, synonyms are expressions 2x + 3x and 5x, since each time the value of the variable x, these expressions acquire the same values \u200b\u200b(this follows from the distribution property of multiplication relative to addition, since 2x + 3x \u003d 5x).

Consider now expressions 3x + 2e and 5h. If x \u003d 1 and B \u003d 1, then the corresponding values \u200b\u200bof these expressions are equal to each other:

3 + 2y \u003d 3 ∙ 1 + 2 ∙ 1 \u003d 5; 5h \u003d 5 ∙ 1 ∙ 1 \u003d 5.

However, it is possible to specify the values \u200b\u200bof x and y, for which the values \u200b\u200bof these expressions will not be equal to each other. For example, if x \u003d 2; y \u003d 0, then

3x + 2y \u003d 3 ∙ 2 + 2 ∙ 0 \u003d 6, 5h \u003d 5 ∙ 20 \u003d 0.

Therefore, there are such values \u200b\u200bof variables in which the corresponding values \u200b\u200bof expressions 3x + 2u and 5h are not equal to each other. Therefore, expressions 3x + 2e and 5h are not identical equal.

Based on the foregoing, identities, in particular, are equality: 2 (x - 1) \u003d 2x - 2 and 2x + 3x \u003d 5x.

The identity is each equality that are written to the well-known properties of actions above the numbers. For example,

a + b \u003d b + a; (A + B) + C \u003d A + (B + C); A (B + C) \u003d AB + AS;

aB \u003d BA; (AB) C \u003d A (BC); A (B - C) \u003d AB - AU.

The identities are such equality:

a + 0 \u003d a; a ∙ 0 \u003d 0; a ∙ (-b) \u003d -ab;

a + (-a) \u003d 0; a ∙ 1 \u003d a; A ∙ (-b) \u003d AB.

1 + 2 + 3 = 6; 5 2 + 12 2 = 13 2 ; 12 ∙ (7 - 6) = 3 ∙ 4.

If in the expression-5x + 2x - 9 to reduce such terms, we obtain that 5x + 2x - 9 \u003d 7x - 9. In this case, it is said that the expression 5x + 2x - 9 was replaced by an identical expression to it 7x - 9.

The identity conversions of expressions with variables are performed using the properties of the number of numbers. In particular, identical transformations with disclosure of brackets, the construction of such terms and the like.

The identical conversions have to be performed when simplifying the expression, that is, the replacement of some expression on the expression identically equal to it, which should be shorter.

Example 1. Simplify the expression:

1) -0.3 m ∙ 5n;

2) 2 (3x - 4) + 3 (-4x + 7);

3) 2 + 5A - (A - 2B) + (3B - a).

1) -0.3 m ∙ 5n \u003d -0.3 ∙ 5mn \u003d -1.5 Mn;

2) 2 (3x 4) + 3 (-4 + 7) \u003d 6 x. - 8 - 1 2x + 21 \u003d 6x + 13;

3) 2 + 5a - (a - 2b) + (3b - a) \u003d 2 + 5A. - but + 2 b. + 3 b. - but \u003d 3A + 5B + 2.

To prove that equality is a identity (in other words, to prove identity, use identical transformations of expressions.

You can prove identity in one of the following ways:

perform identical transformations of its left side, thereby mixing the right part;
perform identical transformations of its right part, thereby minimizing the species of the left side;
to perform identical conversions of both its parts, thereby erecting both parts to the same expressions.

Example 2. Prove identity:

1) 2x - (x + 5) - 11 \u003d x - 16;

2) 206 - 4a \u003d 5 (2a - 3b) - 7 (2a - 5b);

3) 2 (3x - 8) + 4 (5x - 7) \u003d 13 (2x - 5) + 21.

R a s in 'I z and n n.

1) We transform the left part of this equality:

2x - (x + 5) - 11 \u003d 2x - h.- 5 - 11 \u003d x - 16.

The expression on the left part of the equality was identical transformations in the left part of equality, and thereby proved that this equality is a identity.

2) We transform the right side of this equality:

5 (2a - 3b) - 7 (2a - 5b) \u003d 10A. - 15 b. - 14A. + 35 b. \u003d 20b - 4a.

The right-hand side of the equality with the form of the left side of the equality, and thereby proved that this equality is the identity.

3) in this case it is convenient to simplify both the left and right of equality and compare the results:

2 (3 - 8) + 4 (5x - 7) \u003d 6x - 16 + 20x - 28 \u003d 26x - 44;

13 (2x - 5) + 21 \u003d 26x - 65 + 21 \u003d 26x - 44.

The left and right parts of the equality with the same transformations of the left and right side of equality: 26x - 44. Therefore, this equality is a identity.

What expressions are identical? Give an example of identical expressions. What equality is called identity? Give an example of identity. What is called the identical transformation of the expression? How to prove identity?

(Orally) or there are expressions identically equal:

1) 2a + a and 3a;

2) 7x + 6 and 6 + 7x;

3) x + x + x and x 3;

4) 2 (x - 2) and 2x - 4;

5) M - n and n - m;

6) 2a ∙ r and 2r ∙ e?

Whether identically equal expressions:

1) 7x - 2x and 5x;

2) 5a - 4 and 4 - 5a;

3) 4m + n and n + 4m;

4) a + a and a 2;

5) 3 (a - 4) and 3a - 12;

6) 5m ∙ n and 5m + n?

(Orally) is the identity of equality:

1) 2a + 106 \u003d 12AB;

2) 7p - 1 \u003d -1 + 7p;

3) 3 (x - y) \u003d 3x - 5y?

Open parenthesis:

Open parenthesis:

Two similar terms:

Name several expressions, identical expressions 2a + 3a.
Simplify the expression using the rearrangement and connecting properties of multiplication:

1) -2.5 x ∙ 4;

2) 4r ∙ (-1.5);

3) 0.2 x ∙ (0.3 g);

4) - x ∙<-7у).

Simplify the expression:

1) -2p ∙ 3.5;

2) 7a ∙ (-1.2);

3) 0.2 x ∙ (-3u);

4) - 1 m ∙ (-3n).

(Orally) Simplify the expression:

1) 2x - 9 + 5x;

2) 7a - 3b + 2a + 3b;

4) 4a ∙ (-2b).

Two similar terms:

1) 56 - 8A + 4B - A;

2) 17 - 2p + 3r + 19;

3) 1.8 A + 1.9 B + 2.8 A - 2.9 B;

4) 5 - 7C + 1.9 g + 6.9 s - 1.7 g

1) 4 (5x - 7) + 3x + 13;

2) 2 (7 - 9a) - (4 - 18a);

3) 3 (2p - 7) - 2 (g - 3);

4) - (3m - 5) + 2 (3m - 7).

Open brackets and twist similar terms:

1) 3 (8a - 4) + 6a;

2) 7p - 2 (3r - 1);

3) 2 (3x - 8) - 5 (2x + 7);

4) 3 (5m - 7) - (15m - 2).

1) 0.6 x + 0.4 (x - 20), if x \u003d 2.4;

2) 1.3 (2a - 1) - 16.4, if A \u003d 10;

3) 1,2 (M - 5) - 1.8 (10 - m), if M \u003d -3.7;

4) 2x - 3 (x + y) + 4y, if x \u003d -1, y \u003d 1.

Simplify the expression and find its value:

1) 0.7 x + 0.3 (x - 4), if x \u003d -0.7;

2) 1.7 (y - 11) - 16.3, if B \u003d 20;

3) 0.6 (2a - 14) - 0.4 (5a - 1), if A \u003d -1;

4) 5 (m - n) - 4m + 7n, if m \u003d 1.8; N \u003d -0.9.

Prove identity:

1) - (2x - y) \u003d y - 2x;

2) 2 (x - 1) - 2x \u003d -2;

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

4) C - 2 \u003d 5 (C + 2) - 4 (C + 3).

Prove identity:

1) - (M - 3N) \u003d 3N - m;

2) 7 (2 - p) + 7p \u003d 14;

3) 5a \u003d 3 (a - 4) + 2 (a + 6);

4) 4 (M - 3) + 3 (M + 3) \u003d 7M - 3.

The length of one of the sides of the triangle is a cm, and the length of each of the two other sides is 2 cm more than it. Record in the form of an expression perimeter of a triangle and simplify the expression.
The width of the rectangle is x cm, and the length is 3 cm more width. Write down in the form of an expression perimeter of the rectangle and simplify the expression.

1) X - (x - (2x - 3));

2) 5m - ((n - m) + 3N);

3) 4p - (3r - (2p - (r + 1)));

4) 5x - (2x - ((y - x) - 2th));

5) (6a - b) - (4 A - 33B);

6) - (2.7 m - 1.5 n) + (2n - 0.48 m).

Expand brackets and simplify the expression:

1) A - (A - (3A - 1));

2) 12m - ((a - m) + 12a);

3) 5y - (6th - (7th - (8th - 1)));

6) (2.1 A - 2.8 B) - (1A - 1B).

Prove identity:

1) 10x - (- (5x + 20)) \u003d 5 (3x + 4);

2) - (- 3r) - (- (8 - 5p)) \u003d 2 (4 - g);

3) 3 (a - b - c) + 5 (a - b) + 3c \u003d 8 (a - b).

Prove identity:

1) 12a - ((8a - 16)) \u003d -4 (4 - 5a);

2) 4 (x + y -<) + 5(х - t) - 4y - 9(х - t).

Prove that the value of the expression

1.8 (m - 2) + 1.4 (2 - m) + 0.2 (1.7 - 2m) does not depend on the value of the variable.

Prove that if any value variable is the value of the expression

a - (A - (5a + 2)) - 5 (a - 8)

is the same number.

Prove that the sum of three consecutive even numbers is divided by 6.
Prove that if N is a natural number, then the value of expression -2 (2.5 n - 7) + 2 (3N - 6) is even number.

Exercises for repetition

Alloy weighing 1.6 kg contains 15% copper. How many copper kg is contained in this alloy?
How many percent is the number 20 from its:

1) square;

Tourist 2 h walking on foot and 3 h riding a bike. Tourist overcame 56 km. Find, at what speed the tourist rode a bike if she is 12 km / h more for the speed with which he was walking.

Interesting tasks for students of lazy

In the Championship of the City Football, 11 teams participate. Each team plays with others one match. Prove that at any time the competition there is a team that will hold an even number of matches by this moment or has not yet conducted a single one.

In the course of studying algebra, we faced 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 variables and numeric values \u200b\u200bare present in algebraic fractions, arithmetic Actions: Addition, subtraction, multiplication, exercise. The difference between these concepts is that the polynomials are not divided into variable, and in algebraic fractions, the division into a variable can be produced.

And polynomials and algebraic fractions in mathematics are called rational algebraic expressions. But polynomials are integer rational expressions, and algebraic crushing-rational expressions.

It can be obtained from a fractional - rational expression a whole algebraic expression using the identical transformation, which in this case will be the main property of the fraction - the reduction of fractions. Check it in practice:

Example 1.

Perform a conversion: $ \\ \\ FRAC (x ^ 2-4x + 4) (x-2) $

Decision: Convert this fractional rational equation by using the main property of the crushing, i.e. dividing the numerator and denominator on the same number or expression other than $ 0 $.

Immediately this fraction cannot be cut, it is necessary to convert the numerator.

We convert the expression that standing in the knob's numerator, for this we use the square square formula: $ a ^ 2-2ab + B ^ 2 \u003d ((A-B)) ^ $ 2

The fraction has a view

\\ [\\ 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 in the numerator and in the denominator there is a general multiplier - it is an expression of $ x-2 $, which will produce a reduction in 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 cutting, we obtained that the initial fractional rational expression $ \\ Frac (x ^ 2-4x + 4) (x-2) $ was a polynomial $ x-2 $, i.e. whole rational.

Now we will pay attention to the fact that the expressions of $ \\ FRAC (x ^ 2-4x + 4) (x-2) $ and $ x-2 \\ $ are identical to the expressions of $ \\ FRAC (X-2) not at all values \u200b\u200bof the variable, because In order for the fractional rational expression to exist and it was possible to reduce the $ X-2 $ polynomial denominator of the fraction should not be $ 0 $ (as well as the multiplier to which we produce a reduction. In this example, the denominator and the multiplier coincide, but It happens not always).

The values \u200b\u200bof the variable in which the algebraic fraction will exist are called permissible values \u200b\u200bof the variable.

We put the condition for the denomoter of the fraction: $ x-2 ≠ 0 $, then $ x ≠ $ 2.

It means that the expressions of $ \\ FRAC (x ^ 2-4x + 4) (x-2) $ and $ x-2 $ are identical for all values \u200b\u200bof the variable, except $ 2 $.

Definition 1.

Identically equal Expressions are called those that are equal with all valid values \u200b\u200bof the variable.

An identical conversion is any replacement of the original expression on identically equal to it. For such transformations include the implementation of the actions: addition, subtraction, multiplication, making a common factor behind the bracket, bringing algebraic fractions to a common denominator, reduction of algebraic fractions, bringing similar terms, etc. It is necessary to take into account that a number of transformations, such as abbreviation, bringing such terms can change the values \u200b\u200bof the variable.

Receptions used to evict identities

Create the left part of the identity to the right or vice versa using identical transformations

Bring both parts to the same expression using identical transformations.

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

Which of the above techniques to use to prove this identity depends on the initial 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 techniques, namely, we will transform the left side of the identity before its equality with the right.

Consider the left part of the identity: $ \\ ((A + B + C)) ^ 2- 2 (AB + AC + BC) $ - it represents the difference of two polynomials. At the same time, the first polynomial is the square of the sum of the three components. For the construction of 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 multiplication of the number on the polynomial. Alternatively, it is necessary to multiply a common multiplier behind the brackets for each component of the polynomial standing in brackets. When 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) $

We note that in front of the bracket is the sign "-" means when disclosing brackets, all the signs that were changed to the opposite brackets.

$ ((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 $

We will give similar terms, then we get a $ 2ab $, $ 2ac $, $ \\ 2B $ and $ -2ab $, $ - 2ac $ and $ -2bc $, $ 2ac $, $ -2bc $ are mutually destroyed, i.e. Their amount 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 identical transformations, we obtained a 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 initial identity is - short.

We note that in the initial identity, all values \u200b\u200bof the variable are allowed, which means we have proven identity using identical conversions, and it is true for all valid values \u200b\u200bof the variable.