After we have dealt with the concept of identities, we can move on to studying identically equal expressions. The purpose of this article is to explain what it is and show with examples which expressions will be identically equal to others.
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Identically equal expressions: definition
The concept of identically equal expressions is usually studied together with the concept of identity itself within the framework school course algebra. Here is the basic definition taken from one textbook:
Definition 1
Identically equal there will be expressions to each other whose values will be the same for any possible values variables included in their composition.
Also, those numerical expressions to which the same values will correspond are considered identically equal.
This is a fairly broad definition that will be true for all integer expressions whose meaning does not change when the values of the variables change. However, later it becomes necessary to clarify this definition, since in addition to integers, there are other types of expressions that will not make sense with certain variables. This gives rise to the concept of admissibility and inadmissibility of certain variable values, as well as the need to determine the range of permissible values. Let us formulate a refined definition.
Definition 2
Identically equal expressions are those expressions whose values are equal to each other for any acceptable values variables included in their composition. Numerical expressions will be identically equal to each other provided the values are the same.
The phrase “for any valid values of the variables” indicates all those values of the variables for which both expressions will make sense. We will explain this point later when we give examples of identically equal expressions.
You can also provide the following definition:
Definition 3
Identically in equal terms expressions located in the same identity on the left and right sides are called.
Examples of expressions that are identically equal to each other
Using the definitions given above, let's look at a few examples of such expressions.
Let's start with numerical expressions.
Example 1
Thus, 2 + 4 and 4 + 2 will be identically equal to each other, since their results will be equal (6 and 6).
Example 2
In the same way, the expressions 3 and 30 are identically equal: 10, (2 2) 3 and 2 6 (to calculate the value of the last expression you need to know the properties of the degree).
Example 3
But the expressions 4 - 2 and 9 - 1 will not be equal, since their values are different.
Let's move on to examples of literal expressions. a + b and b + a will be identically equal, and this does not depend on the values of the variables (the equality of expressions in this case is determined by the commutative property of addition).
Example 4
For example, if a is equal to 4 and b is equal to 5, then the results will still be the same.
Another example of identically equal expressions with letters is 0 · x · y · z and 0 . Whatever the values of the variables in this case, when multiplied by 0, they will give 0. The unequal expressions are 6 · x and 8 · x, since they will not be equal for any x.
In the event that the areas of permissible values of the variables coincide, for example, in the expressions a + 6 and 6 + a or a · b · 0 and 0, or x 4 and x, and the values of the expressions themselves are equal for any variables, then such expressions are considered identically equal. So, a + 8 = 8 + a for any value of a, and a · b · 0 = 0 too, since multiplying any number by 0 results in 0. The expressions x 4 and x will be identically equal for any x from the interval [ 0 , + ∞) .
But the range of valid values in one expression may be different from the range of another.
Example 5
For example, let's take two expressions: x − 1 and x - 1 · x x. For the first of them, the range of permissible values of x will be the entire set of real numbers, and for the second - the set of all real numbers, with the exception of zero, because then we will get 0 in the denominator, and such a division is not defined. These two expressions have a common range of values formed by the intersection of two separate ranges. We can conclude that both expressions x - 1 · x x and x − 1 will make sense for any real values of the variables, with the exception of 0.
The basic property of the fraction also allows us to conclude that x - 1 · x x and x − 1 will be equal for any x that is not 0. This means that on the general range of permissible values these expressions will be identically equal to each other, but for any real x we cannot speak of identical equality.
If we replace one expression with another, which is identically equal to it, then this process is called an identity transformation. This concept is very important, and we will talk about it in detail in a separate material.
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The numbers and expressions that make up the original expression can be replaced by identically equal expressions. Such a transformation of the original expression leads to an expression that is identically equal to it.
For example, in the expression 3+x, the number 3 can be replaced by the sum 1+2, which will result in the expression (1+2)+x, which is identically equal to the original expression. Another example: in the expression 1+a 5, the power a 5 can be replaced by an identically equal product, for example, of the form a·a 4. This will give us the expression 1+a·a 4 .
This transformation is undoubtedly artificial, and is usually a preparation for some further transformations. For example, in the sum 4 x 3 +2 x 2, taking into account the properties of the degree, the term 4 x 3 can be represented as a product 2 x 2 2 x. After this transformation, the original expression will take the form 2 x 2 2 x+2 x 2. Obviously, the terms in the resulting sum have a common factor of 2 x 2, so we can perform the following transformation - bracketing. After it we come to the expression: 2 x 2 (2 x+1) .
Adding and subtracting the same number
Another artificial transformation of an expression is the addition and simultaneous subtraction of the same number or expression. This transformation is identical because it is essentially equivalent to adding zero, and adding zero does not change the value.
Let's look at an example. Let's take the expression x 2 +2·x. If you add one to it and subtract one, this will allow you to perform another identical transformation in the future - square the binomial: x 2 +2 x=x 2 +2 x+1−1=(x+1) 2 −1.
Bibliography.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Mordkovich A. G. Algebra. 7th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
Identity transformations represent the work we do with numbers and literal expressions, as well as with 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 identity transformations in this topic.
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Identical transformation of an expression. What it is?
We first encountered the concept of identical transformed in algebra lessons in the 7th grade. It was then that we first became 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 expression transformation– these are actions performed with the aim of replacing the original expression with an expression that will be identically equal to the original one.
Often this definition is used in an abbreviated form, in which the word “identical” is omitted. It is assumed that in any case we transform the expression in such a way as to obtain an expression identical to the original one, and this does not need to be emphasized separately.
Let us illustrate this definition with examples.
Example 1
If we replace the expression x + 3 − 2 to an identically equal expression x+1, then we will carry out an identical transformation of the expression x + 3 − 2.
Example 2
Replacing the expression 2 a 6 with the expression a 3 is an identity transformation, whereas replacing the expression x to the expression x 2 is not an identity 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. Usually we write the original and the resulting expression as an equality. Thus, writing x + 1 + 2 = x + 3 means that the expression x + 1 + 2 has been reduced to the form x + 3.
Consecutive execution of actions leads us to a chain of equalities, which represents several identical transformations located in a row. Thus, we understand the entry x + 1 + 2 = x + 3 = 3 + x as the sequential implementation of two transformations: first, the expression x + 1 + 2 was brought to the form x + 3, and it was brought to the form 3 + x.
Identical transformations and ODZ
A number of expressions that we begin to study in 8th grade do not make sense for all values of the variables. Carrying out identical transformations in these cases requires us to pay attention to the range of permissible values of variables (APV). Performing identical transformations can leave the ODZ unchanged or narrow it.
Example 3
When performing a transition from an expression a + (− b) to the expression a − b range of permissible variable values a And b remains the same.
Example 4
Moving from expression x to expression x 2 x leads to a narrowing of the range of permissible values of the variable x from the set of all real numbers to the set of all real numbers, from which zero has been excluded.
Example 5
Identical expression transformation x 2 x expression x leads to an expansion of the range of permissible values of 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 of variables when carrying out identity transformations is important when solving problems, since it can affect the accuracy of calculations and lead to errors.
Basic identity transformations
Let's now see what identity transformations are and how they are performed. Let us single out those types of identity transformations that we deal with most often into a group of basic ones.
In addition to the main identity transformations, there are a number of transformations that relate to expressions of a specific type. For fractions, these are techniques for reducing and bringing 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 carried out based on the properties of logarithms. For trigonometric expressions, all operations using trigonometric formulas. All these particular transformations are discussed in detail 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 consider the main identity transformations.
Rearranging terms and factors
Let's start by rearranging the terms. We deal with this identical transformation most often. And the main rule here can be considered the following statement: in any sum, rearranging the terms does not affect the result.
This rule is based on the commutative and associative properties of addition. These properties allow us to rearrange terms and obtain expressions that are identically equal to the original ones. That is why rearranging the terms in the sum is an identity transformation.
Example 6
We have the sum of three terms 3 + 5 + 7. If we swap terms 3 and 5, then the expression will take the form 5 + 3 + 7. There are several options for swapping terms in this case. All of them lead to expressions identically equal to the original one.
Not only numbers, but also expressions can act as terms in the sum. They, just like numbers, can be rearranged without affecting the final result of the calculations.
Example 7
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, like this (- 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 a sum in which the terms can be swapped.
Just like terms, you can swap factors in the original expressions and obtain identically correct equations. This action is governed by the following rule:
Definition 2
In a product, rearranging factors does not affect the result of calculations.
This rule is based on the commutative and combinative properties of multiplication, which confirm the correctness of the identical transformation.
Example 8
Work 3 5 7 by rearranging the 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 parentheses
Parentheses can contain numeric and variable expressions. These expressions can be transformed into identically equal expressions, in which there will be no parentheses at all or fewer of them than in the original expressions. This method of transforming expressions is called parenthesis expansion.
Example 10
Let's carry out operations with brackets in an expression of the form 3 + x − 1 x in order to obtain the identically correct expression 3 + x − 1 x.
The expression 3 x - 1 + - 1 + x 1 - x can be transformed into the identically equal expression without parentheses 3 x - 3 - 1 + x 1 - x.
We discussed in detail the rules for converting expressions with brackets in the topic “Expanding brackets,” which is posted on our resource.
Grouping of terms, factors
In cases where we are dealing with three or more terms, we can resort to this type of identity transformations as grouping terms. This method of transformation means combining several terms into a group by rearranging them and putting them in brackets.
When grouping, the terms are swapped so that the grouped terms are side by side in the expression record. 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 either by simple numbers or by expressions. Grouping rules were discussed in detail in the topic “Grouping Addends and Factors.”
Replacing differences with sums, partial products and vice versa
Replacing differences with sums became possible thanks to our familiarity with opposite numbers. Now subtracting from a number a numbers b can be considered as an addition to a number a numbers − b. Equality a − b = a + (− b) can be considered fair and, on its basis, replace differences with sums.
Example 13
Let's take the expression 4 + 3 − 2 , in which the difference of numbers 3 − 2 we can write it 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 proceed to sums from any differences. Similarly, we can make the reverse replacement.
Replacing division with multiplication with the reciprocal of the divisor becomes possible thanks to the concept of reciprocal numbers. This transformation can be written as a: b = 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 by x can be multiplied by 1 x. Division by x+3 we can replace by multiplying by 1 x + 3. The transformation allows us to obtain an expression identical to the original: 1 + 5 · 1 x · 1 x + 3.
Replacing multiplication by division is carried out according to the scheme a · b = 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.
Doing things with numbers
Performing operations with numbers is subject to the rule of the order in which actions are performed. First, operations are carried out 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 you can carry out all other actions from left to right. It is important to remember that multiplication and division come before addition and subtraction.
Operations with numbers allow you to transform the original expression into an identical one equal to it.
Example 18
Let's transform the expression 3 · 2 3 - 1 · a + 4 · x 2 + 5 · x by performing all possible operations with numbers.
Solution
First of all, let's pay attention to the degree 2 3 and root 4 and calculate their values: 2 3 = 8 and 4 = 2 2 = 2 .
Let's substitute the obtained values into the original expression and get: 3 · (8 - 1) · a + 2 · (x 2 + 5 · x) .
Now let's do the steps in brackets: 8 − 1 = 7 . And let's move on to the expression 3 · 7 · a + 2 · (x 2 + 5 · x) .
All we have to do is multiply numbers 3 And 7 . We get: 21 · a + 2 · (x 2 + 5 · x) .
Answer: 3 2 3 - 1 a + 4 x 2 + 5 x = 21 a + 2 (x 2 + 5 x)
Operations with numbers may be preceded by other types of identity transformations, such as grouping numbers or opening parentheses.
Example 19
Let's take the expression 3 + 2 (6:3) x (y 3 4) − 2 + 11.
Solution
First of all, we will replace the quotient in brackets 6: 3 on its meaning 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 = 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 do the steps in brackets: (3 − 2 + 11) + (2 2 4) x y 3 = 12 + 16 x y 3
Answer:3 + 2 (6:3) x (y 3 4) − 2 + 11 = 12 + 16 x y 3
If we work with numerical expressions, then the goal of our work will be to find the value of the expression. If we transform expressions with variables, then the goal of our actions will be to simplify the expression.
Bracketing out the common factor
In cases where the terms in the expression have the same factor, we can take this common factor out of brackets. To do this, we first need to represent the original expression as the product of a common factor and an expression in parentheses, which consists of the original terms without a common factor.
Example 20
Numerically 2 7 + 2 3 we can take out the common factor 2 outside the brackets and obtain an identically correct expression of the form 2 (7 + 3).
You can refresh your memory of the rules for putting the common factor out of brackets in the corresponding section of our resource. The material discusses in detail the rules for taking the common factor out of brackets and provides numerous examples.
Reducing similar terms
Now let's move on to sums that contain similar terms. There are two options here: sums containing identical terms, and sums whose terms differ by a numerical coefficient. Operations with sums containing similar terms are called reduction of similar terms. It is carried out as follows: we take the common letter part out of brackets and calculate the sum of the numerical coefficients in brackets.
Example 21
Consider the expression 1 + 4 x − 2 x. We can take the literal part x out of brackets and get the expression 1 + x (4 − 2). Let's calculate the value of the expression in brackets and get a sum of the form 1 + x · 2.
Replacing numbers and expressions with identically equal expressions
The numbers and expressions that make up the original expression can be replaced by identically equal expressions. Such a transformation of the original expression leads to an expression that is identically equal to it.
Example 22 Example 23
Consider the expression 1 + a 5, in which we can replace the degree a 5 with a product identically equal to it, 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 changes.
Example 24
Consider the transformation of the sum 4 x 3 + 2 x 2. Here the term 4 x 3 we can imagine 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 isolate the common factor 2 x 2 and put it out of brackets: 2 x 2 (2 x + 1).
Adding and subtracting 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 subsequently carry out another identical transformation - to isolate the square of the binomial: x 2 + 2 x = x 2 + 2 x + 1 − 1 = (x + 1) 2 − 1.
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Having gained an idea of identities, it is logical to move on to getting acquainted with. In this article we will answer the question of what identically equal expressions are, and also use examples to understand which expressions are identically equal and which are not.
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What are identically equal expressions?
The definition of identically equal expressions is given in parallel with the definition of identity. This happens in 7th grade algebra class. In the textbook on algebra for 7th grade by the author Yu. N. Makarychev, the following formulation is given:
Definition.
– these are expressions whose values are equal for any values of the variables included in them. Numeric expressions that answer same values, also called identically equal.
This definition is used up to grade 8; it is valid for integer expressions, since they make sense for any values of the variables included in them. And in grade 8, the definition of identically equal expressions is clarified. Let us explain what this is connected with.
In the 8th grade, the study of other types of expressions begins, which, unlike whole expressions, may not make sense for some values of the variables. This forces us to introduce definitions of permissible and unacceptable values of variables, as well as the range of permissible values of the variable’s variable value, and, as a consequence, to clarify the definition of identically equal expressions.
Definition.
Two expressions whose values are equal for all permissible values of the variables included in them are called identically equal expressions. Two numerical expressions having the same values are also called identically equal.
IN this definition identically equal expressions, it is worth clarifying the meaning of the phrase “for all permissible values of the variables included in them.” It implies all such values of variables for which both identically equal expressions make sense at the same time. We will explain this idea in the next paragraph by looking at examples.
The definition of identically equal expressions in A. G. Mordkovich’s textbook is given a little differently:
Definition.
Identically equal expressions– these are expressions on the left and right sides of the identity.
The meaning of this and the previous definitions coincide.
Examples of identically equal expressions
The definitions introduced in the previous paragraph allow us to give examples of identically equal expressions.
Let's start with identically equal numerical expressions. The numerical expressions 1+2 and 2+1 are identically equal, since they correspond to equal values 3 and 3. The expressions 5 and 30:6 are also identically equal, as are the expressions (2 2) 3 and 2 6 (the values of the latter expressions are equal by virtue of ). But the numerical expressions 3+2 and 3−2 are not identically equal, since they correspond to the values 5 and 1, respectively, and they are not equal.
Now let's give examples of identically equal expressions with variables. These are the expressions a+b and b+a. Indeed, for any values of the variables a and b, the written expressions take the same values (as follows from the numbers). For example, with a=1 and b=2 we have a+b=1+2=3 and b+a=2+1=3 . For any other values of the variables a and b, we will also obtain equal values of these expressions. The expressions 0·x·y·z and 0 are also identically equal for any values of the variables x, y and z. But the expressions 2 x and 3 x are not identically equal, since, for example, when x=1 their values are not equal. Indeed, for x=1, the expression 2 x is equal to 2 x 1=2, and the expression 3 x is equal to 3 x 1=3.
When the ranges of permissible values of variables in expressions coincide, as, for example, in the expressions a+1 and 1+a, or a·b·0 and 0, or and, and the values of these expressions are equal for all values of the variables from these areas, then here everything is clear - these expressions are identically equal for all permissible values of the variables included in them. So a+1≡1+a for any a, the expressions a·b·0 and 0 are identically equal for any values of the variables a and b, and the expressions and are identically equal for all x of ; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
While studying algebra, we came across the concepts of a polynomial (for example ($y-x$,$\ 2x^2-2x$, etc.) and algebraic fraction (for example $\frac(x+5)(x)$, $\frac(2x ^2)(2x^2-2x)$,$\ \frac(x-y)(y-x)$, etc.) The similarity of these concepts is that both polynomials and algebraic fractions contain variables and numeric values, arithmetic operations are performed: addition, subtraction, multiplication, exponentiation. The difference between these concepts is that in polynomials division by a variable is not performed, but in algebraic fractions division by a variable can be performed.
Both polynomials and algebraic fractions are called rational algebraic expressions in mathematics. But polynomials are whole rational expressions, and algebraic fractions fractional-rational expressions.
Can be obtained from fractionally --rational expression whole algebraic expression using an identity transformation, which in this case will be the main property of a fraction - reduction of fractions. Let's check this in practice:
Example 1
Convert:$\ \frac(x^2-4x+4)(x-2)$
Solution: Convert given fractional rational equation is possible by using the basic property of the reduction fraction, i.e. dividing the numerator and denominator by the same number or expression other than $0$.
This fraction cannot be reduced immediately; the numerator must be transformed.
Let's transform the expression in the numerator of the fraction, for this we use the formula for the square of the difference: $a^2-2ab+b^2=((a-b))^2$
The fraction looks like
\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)\]
Now we see that the numerator and denominator have a common factor - this is the expression $x-2$, by which we will reduce the fraction
\[\frac(x^2-4x+4)(x-2)=\frac(x^2-4x+4)(x-2)=\frac(((x-2))^2)( x-2)=\frac(\left(x-2\right)(x-2))(x-2)=x-2\]
After reduction, we found that the original fractional rational expression $\frac(x^2-4x+4)(x-2)$ became a polynomial $x-2$, i.e. whole rational.
Now let us 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 of the variable, because In order for a fractional rational expression to exist and to be able 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 are reducing. In in this example the denominator and the multiplier are the same, but this is not always the case).
The values of the variable at which the algebraic fraction will exist are called the permissible values of the variable.
Let's put a condition on the denominator of the fraction: $x-2≠0$, then $x≠2$.
This means that the expressions $\frac(x^2-4x+4)(x-2)$ and $x-2$ are identical for all values of the variable except $2$.
Definition 1
Identically equal expressions are those that are equal for all valid values of the variable.
An identical transformation is any replacement of the original expression with an identically equal one. Such transformations include performing actions: addition, subtraction, multiplication, placing a common factor out of brackets, reduction algebraic fractions to a common denominator, reducing algebraic fractions, reducing similar terms, etc. It is necessary to take into account that a number of transformations, such as reduction, reduction of similar terms, can change the permissible values of the variable.
Techniques used to prove identities
Bring the left side of the identity to the right or vice versa using identity transformations
Reduce both sides to the same expression using identical transformations
Transfer the expressions in one part of the expression to another and prove that the resulting difference is equal to $0$
Which of the above methods to use to prove a given identity depends on the original identity.
Example 2
Prove the identity $((a+b+c))^2- 2(ab+ac+bc)=a^2+b^2+c^2$
Solution: To prove this identity, we use the first of the above methods, namely, we will transform the left side of the identity until it is equal to the right.
Let's consider the left side of the identity: $\ ((a+b+c))^2- 2(ab+ac+bc)$ - it represents the difference of two polynomials. In this case, the first polynomial is the square of the sum of three terms. To square the sum of several terms, we use the formula:
\[((a+b+c))^2=a^2+b^2+c^2+2ab+2ac+2bc\]
To do this, we need to multiply a number by a polynomial. Remember that for this we need to multiply the common factor behind the brackets by each term of the polynomial in the brackets. Then we get:
$2(ab+ac+bc)=2ab+2ac+2bc$
Now let's return to the original polynomial, it will take the form:
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)$
Please note that before the bracket there is a “-” sign, which means that when the brackets are opened, all the signs that were in the brackets change to the opposite.
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc$
Let us present similar terms, then we obtain that the monomials $2ab$, $2ac$,$\ 2bc$ and $-2ab$,$-2ac$, $-2bc$ cancel each other out, i.e. their sum is $0$.
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2+2ab+2ac+2bc-(2ab+2ac+2bc)= a ^2+b^2+c^2+2ab+2ac+2bc-2ab-2ac-2bc=a^2+b^2+c^2$
This means that by means of identical transformations we have obtained an identical expression on the left side of the original identity
$((a+b+c))^2- 2(ab+ac+bc)=\ a^2+b^2+c^2$
Note that the resulting expression shows that the original identity is true.
Please note that in the original identity all values of the variable are valid, which means we proved the identity using identity transformations, and it is true for all valid values of the variable.