This article is about transformation of rational expressions, mostly fractionally rational, is one of the key questions of the algebra course for grades 8. First, we recall what kind of expressions are called rational. Next, we will focus on performing standard transformations with rational expressions, such as grouping terms, taking common factors out of brackets, reducing similar terms, etc. Finally, we will learn how to represent fractional rational expressions as rational fractions.

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Definition and examples of rational expressions

Rational expressions are one of the types of expressions studied in algebra lessons at school. Let's give a definition.

Definition.

Expressions made up of numbers, variables, brackets, degrees with integer exponents, connected using the signs of arithmetic operations +, −, · and:, where division can be indicated by a fraction bar, are called rational expressions.

Here are some examples of rational expressions: .

Rational expressions begin to be purposefully studied in the 7th grade. Moreover, in the 7th grade, the basics of working with the so-called whole rational expressions, that is, with rational expressions that do not contain division into expressions with variables. To do this, monomials and polynomials are consistently studied, as well as the principles for performing actions with them. All this knowledge eventually allows you to perform the transformation of integer expressions.

In grade 8, they move on to the study of rational expressions containing division by an expression with variables, which are called fractional rational expressions. In this regard, special attention is paid to the so-called rational fractions(also called algebraic fractions), that is, fractions whose numerator and denominator contain polynomials. This ultimately makes it possible to perform the transformation of rational fractions.

The acquired skills allow us to proceed to the transformation of rational expressions of an arbitrary form. This is explained by the fact that any rational expression can be considered as an expression composed of rational fractions and integer expressions, connected by signs of arithmetic operations. And we already know how to work with integer expressions and algebraic fractions.

The main types of transformations of rational expressions

With rational expressions, you can carry out any of the basic identity transformations, whether it is a grouping of terms or factors, bringing like terms, performing operations with numbers, etc. Typically, the purpose of these transformations is rational expression simplification.

Example.

.

Solution.

It is clear that this rational expression is the difference of two expressions and , moreover, these expressions are similar, since they have the same literal part. Thus, we can perform a reduction of like terms:

Answer:

.

It is clear that when carrying out transformations with rational expressions, as, indeed, with any other expressions, one must remain within the framework of the accepted order of actions.

Example.

Transform rational expression .

Solution.

We know that the actions in parentheses are executed first. Therefore, first of all, we transform the expression in brackets: 3 x − x=2 x .

Now you can substitute the result in the original rational expression: . So we came to an expression containing the actions of one stage - addition and multiplication.

Let's get rid of the parentheses at the end of the expression by applying the division-by-product property: .

Finally, we can group numeric factors and factors with variable x, and then perform the corresponding operations on numbers and apply : .

This completes the transformation of the rational expression, and as a result we got a monomial.

Answer:

Example.

Transform Rational Expression .

Solution.

First we convert the numerator and denominator. This order of transformation of fractions is explained by the fact that the stroke of a fraction is, in essence, another division designation, and the original rational expression is essentially a particular form , and the actions in parentheses are executed first.

So, in the numerator we perform operations with polynomials, first multiplication, then subtraction, and in the denominator we group the numerical factors and calculate their product: .

Let's also imagine the numerator and denominator of the resulting fraction as a product: suddenly it is possible to reduce the algebraic fraction. To do this, in the numerator we use difference of squares formula, and in the denominator we take the deuce out of brackets, we have .

Answer:

.

So, the initial acquaintance with the transformation of rational expressions can be considered accomplished. We pass, so to speak, to the sweetest.

Representation as a rational fraction

The most common end goal of transforming expressions is to simplify their form. In this light, the simplest form to which a fractionally rational expression can be converted is a rational (algebraic) fraction, and in a particular case, a polynomial, a monomial, or a number.

Is it possible to represent any rational expression in the form rational fraction? The answer is yes. Let's explain why this is so.

As we have already said, any rational expression can be considered as polynomials and rational fractions connected by plus, minus signs, multiply and divide. All relevant operations on polynomials yield a polynomial or a rational fraction. In turn, any polynomial can be converted into an algebraic fraction by writing it with a denominator 1. And addition, subtraction, multiplication and division of rational fractions result in a new rational fraction. Therefore, after performing all the operations with polynomials and rational fractions in a rational expression, we get a rational fraction.

Example.

Express as a rational fraction the expression .

Solution.

The original rational expression is the difference between a fraction and a product of fractions of the form . According to the order of operations, we must first perform the multiplication, and only then the addition.

We start by multiplying algebraic fractions:

We substitute the result obtained into the original rational expression: .

We have come to the subtraction of algebraic fractions with different denominators:

So, having performed actions with rational fractions that make up the original rational expression, we presented it as a rational fraction.

Answer:

.

To consolidate the material, we will analyze the solution of another example.

Example.

Express a rational expression as a rational fraction.

Any fractional expression(item 48) can be written as , where P and Q are rational expressions, and Q necessarily contains variables. Such a fraction is called a rational fraction.

Examples of rational fractions:

The main property of a fraction is expressed by an identity that is valid under the conditions here - a whole rational expression. This means that the numerator and denominator of a rational fraction can be multiplied or divided by the same non-zero number, monomial or polynomial.

For example, the property of a fraction can be used to change the signs of the members of a fraction. If the numerator and denominator of a fraction are multiplied by -1, we get Thus, the value of the fraction will not change if the signs of the numerator and denominator are changed at the same time. If you change the sign of only the numerator or only the denominator, then the fraction will change its sign:

For example,

60. Reduction of rational fractions.

To reduce a fraction is to divide the numerator and denominator of a fraction by a common factor. The possibility of such a reduction is due to the main property of the fraction.

To reduce a rational fraction, you need to factorize the numerator and denominator. If it turns out that the numerator and denominator have common factors, then the fraction can be reduced. If there are no common factors, then the conversion of the fraction by reduction is impossible.

Example. Reduce fraction

Solution. We have

The reduction of the fraction is performed under the condition .

61. Bringing rational fractions to a common denominator.

The common denominator of several rational fractions is the whole rational expression, which is divided by the denominator of each fraction (see item 54).

For example, a polynomial serves as a common denominator of fractions, since it is divisible by and by and by and by a polynomial and a polynomial and a polynomial, etc. Usually such a common denominator is taken that any other common denominator is divisible by Echosen. This simplest denominator is sometimes called the least common denominator.

In the example above, the common denominator is We have

Bringing these fractions to common denominator achieved by multiplying the numerator and denominator of the first fraction by 2. and the numerator and denominator of the second fraction by Polynomials are called additional factors for the first and second fractions, respectively. The additional factor for a given fraction is equal to the quotient of dividing the common denominator by the denominator of the given fraction.

To reduce several rational fractions to a common denominator, you need:

1) decompose the denominator of each fraction into factors;

2) make a common denominator, including as factors in it all the factors obtained in paragraph 1) of the expansions; if a certain factor exists in several expansions, then it is taken with an exponent equal to the largest of the available ones;

3) finding additional factors for each of the fractions (for this, the common denominator is divided by the denominator of the fraction);

4) multiplying the numerator and denominator of each fraction by an additional factor, bring the fraction to a common denominator.

Example. Reduce to a common denominator of a fraction

Solution. Let's factorize the denominators:

The following factors must be included in the common denominator: and the least common multiple of the numbers 12, 18, 24, i.e. . So the common denominator is

Additional multipliers: for the first fraction for the second for the third So, we get:

62. Addition and subtraction of rational fractions.

The sum of two (and in general any finite number) rational fractions with same denominators identically equal to a fraction with the same denominator and numerator, equal to the sum numerators of added fractions:

The situation is similar when subtracting fractions with the same denominators:

Example 1: Simplify an expression

Solution.

To add or subtract rational fractions with different denominators, you must first bring the fractions to a common denominator, and then perform operations on the resulting fractions with the same denominators.

Example 2: Simplify an expression

Solution. We have

63. Multiplication and division of rational fractions.

The product of two (and in general any finite number) rational fractions is identically equal to a fraction whose numerator is equal to the product of the numerators, and the denominator is the product of the denominators of the multiplied fractions:

The quotient of dividing two rational fractions is identically equal to a fraction whose numerator is equal to the product of the numerator of the first fraction by the denominator of the second fraction, and the denominator is the product of the denominator of the first fraction by the numerator of the second fraction:

The formulated rules for multiplication and division also apply to the case of multiplication or division by a polynomial: it is enough to write this polynomial as a fraction with a denominator of 1.

Given the possibility of reducing a rational fraction obtained by multiplying or dividing rational fractions, it is usually sought to factorize the numerators and denominators of the original fractions before performing these operations.

Example 1. Multiply

Solution. We have

Using the multiplication rule for fractions, we get:

Example 2: Perform division

Solution. We have

Using the division rule, we get:

64. Raising a rational fraction to an integer power.

To raise a rational fraction natural degree, you need to raise the numerator and denominator of the fraction to this power separately; the first expression is the numerator and the second expression is the denominator of the result:

Example 1. Convert to a fraction a power of 3.

Solution Solution.

When raising a fraction to a negative integer power, an identity is used that is valid for all values ​​of the variables for which .

Example 2. Convert expression to fraction

65. Transformation of rational expressions.

The transformation of any rational expression comes down to adding, subtracting, multiplying and dividing rational fractions, as well as raising a fraction to a natural power. Any rational expression can be converted into a fraction whose numerator and denominator are integer rational expressions; this is usually the goal identical transformations rational expressions.

Example. Simplify Expression

66. The simplest transformations of arithmetic roots (radicals).

When converting arithmetic coria, their properties are used (see item 35).

Let's consider several examples of applying the properties of arithmetic roots for the simplest transformations of radicals. In this case, all variables will be considered as taking only non-negative values.

Example 1. Extract the root of the product

Solution. Applying property 1°, we get:

Example 2. Take out the factor from under the root sign

Solution.

Such a transformation is called factoring out from under the root sign. The purpose of the transformation is to simplify the radical expression.

Example 3: Simplify.

Solution. According to property 3°, we usually try to simplify the radical expression, for which they take out the multipliers beyond the corium sign. We have

Example 4: Simplify

Solution. We transform the expression by introducing a factor under the sign of the root: By property 4° we have

Example 5: Simplify

Solution. By property 5°, we have the right to divide the exponent of the root and the exponent of the root expression into the same natural number. If in the example under consideration we divide the indicated indicators by 3, then we get .

Example 6. Simplify expressions:

Solution, a) By property 1°, we obtain that to multiply roots of the same degree, it suffices to multiply the root expressions and extract the root of the same degree from the result obtained. Means,

b) First of all, we must reduce the radicals to one index. According to property 5°, we can multiply the exponent of the root by the same natural number. Therefore, Next, we now have in the result obtained by dividing the indicators of the root and the degree of the radical expression by 3, we get .

The article tells about the transformation of rational expressions. Consider the types of rational expressions, their transformations, groupings, bracketing the common factor. Let's learn how to represent fractional rational expressions as rational fractions.

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Definition and examples of rational expressions

Definition 1

Expressions that are made up of numbers, variables, brackets, degrees with the operations of addition, subtraction, multiplication, division with the presence of a fraction bar are called rational expressions.

For example, we have that 5 , 2 3 x - 5 , - 3 a b 3 - 1 c 2 + 4 a 2 + b 2 1 + a: (1 - b) , (x + 1) (y - 2) x 5 - 5 x y 2 - 1 11 x 3 .

That is, these are expressions that do not have division into expressions with variables. The study of rational expressions begins with grade 8, where they are called fractional rational expressions. Particular attention is paid to fractions in the numerator, which are converted using transformation rules.

This allows us to proceed to the transformation of rational fractions of an arbitrary form. Such an expression can be considered as an expression with the presence of rational fractions and integer expressions with action signs.

The main types of transformations of rational expressions

Rational expressions are used to perform identical transformations, groupings, casting like ones, and performing other operations with numbers. The purpose of such expressions is to simplify.

Example 1

Convert rational expression 3 · x x · y - 1 - 2 · x x · y - 1 .

Solution

It can be seen that such a rational expression is the difference 3 · x x · y - 1 and 2 · x x · y - 1 . Notice that they have the same denominator. This means that the reduction of similar terms takes the form

3 x x y - 1 - 2 x x y - 1 = x x y - 1 3 - 2 = x x y - 1

Answer: 3 x x y - 1 - 2 x x y - 1 = x x y - 1 .

Example 2

Perform the transformation 2 · x · y 4 · (- 4) · x 2: (3 · x - x) .

Solution

Initially, we perform actions in brackets 3 · x − x = 2 · x . This expression represent in the form 2 x y 4 (- 4) x 2: (3 x - x) \u003d 2 x y 4 (- 4) x 2: 2 x. We arrive at an expression that contains actions with one stage, that is, it has addition and subtraction.

Get rid of parentheses by using the division property. Then we get that 2 x y 4 (- 4) x 2: 2 x = 2 x y 4 (- 4) x 2: 2: x .

We group the numerical factors with the variable x, after that we can perform operations with powers. We get that

2 x y 4 (- 4) x 2: 2: x = (2 (- 4) : 2) (x x 2: x) y 4 = - 4 x 2 y 4

Answer: 2 x y 4 (- 4) x 2: (3 x - x) = - 4 x 2 y 4 .

Example 3

Convert an expression of the form x · (x + 3) - (3 · x + 1) 1 2 · x · 4 + 2 .

Solution

First, let's convert the numerator and denominator. Then we get an expression of the form (x · (x + 3) - (3 · x + 1)) : 1 2 · x · 4 + 2, and the actions in brackets are done first. In the numerator, actions are performed and factors are grouped. Then we get an expression of the form x (x + 3) - (3 x + 1) 1 2 x 4 + 2 = x 2 + 3 x - 3 x - 1 1 2 4 x + 2 = x 2 - 1 2 x + 2 .

We transform the formula for the difference of squares in the numerator, then we get that

x 2 - 1 2 x + 2 = (x - 1) (x + 1) 2 (x + 1) = x - 1 2

Answer: x (x + 3) - (3 x + 1) 1 2 x 4 + 2 = x - 1 2 .

Representation as a rational fraction

An algebraic fraction is most often subjected to simplification when solving. Each rational is brought to this in different ways. It is necessary to perform all the necessary operations with polynomials so that the rational expression can eventually give a rational fraction.

Example 4

Express as a rational fraction a + 5 a (a - 3) - a 2 - 25 a + 3 1 a 2 + 5 a .

Solution

This expression can be represented as a 2 - 25 a + 3 1 a 2 + 5 a . Multiplication is performed first of all according to the rules.

We should start with multiplication, then we get that

a 2 - 25 a + 3 1 a 2 + 5 a = a - 5 (a + 5) a + 3 1 a (a + 5) = a - 5 (a + 5) 1 ( a + 3) a (a + 5) = a - 5 (a + 3) a

We produce a representation of the result obtained with the original. We get that

a + 5 a (a - 3) - a 2 - 25 a + 3 1 a 2 + 5 a = a + 5 a a - 3 - a - 5 a + 3 a

Now let's do the subtraction:

a + 5 a a - 3 - a - 5 a + 3 a = a + 5 a + 3 a (a - 3) (a + 3) - (a - 5) (a - 3) (a + 3) a (a - 3) = = a + 5 a + 3 - (a - 5) (a - 3) a (a - 3) (a + 3) = a 2 + 3 a + 5 a + 15 - (a 2 - 3 a - 5 a + 15) a (a - 3) (a + 3) = = 16 a a (a - 3) (a + 3) = 16 a - 3 (a + 3) = 16 a 2 - 9

After that, it is obvious that the original expression will take the form 16 a 2 - 9 .

Answer: a + 5 a (a - 3) - a 2 - 25 a + 3 1 a 2 + 5 a = 16 a 2 - 9 .

Example 5

Express x x + 1 + 1 2 x - 1 1 + x as a rational fraction.

Solution

The given expression is written as a fraction, in the numerator of which there is x x + 1 + 1, and in the denominator 2 x - 1 1 + x. It is necessary to make transformations x x + 1 + 1 . To do this, you need to add a fraction and a number. We get that x x + 1 + 1 = x x + 1 + 1 1 = x x + 1 + 1 (x + 1) 1 (x + 1) = x x + 1 + x + 1 x + 1 = x + x + 1 x + 1 = 2 x + 1 x + 1

It follows that x x + 1 + 1 2 x - 1 1 + x = 2 x + 1 x + 1 2 x - 1 1 + x

The resulting fraction can be written as 2 x + 1 x + 1: 2 x - 1 1 + x .

After division, we arrive at a rational fraction of the form

2 x + 1 x + 1: 2 x - 1 1 + x = 2 x + 1 x + 1 1 + x 2 x - 1 = 2 x + 1 (1 + x) (x + 1) (2 x - 1) = 2 x + 1 2 x - 1

You can solve it differently.

Instead of dividing by 2 x - 1 1 + x, we multiply by the reciprocal of 1 + x 2 x - 1 . Applying the distribution property, we get that

x x + 1 + 1 2 x - 1 1 + x = x x + 1 + 1: 2 x - 1 1 + x = x x + 1 + 1 1 + x 2 x - 1 = = x x + 1 1 + x 2 x - 1 + 1 1 + x 2 x - 1 = x 1 + x (x + 1) 2 x - 1 + 1 + x 2 x - 1 = = x 2 x - 1 + 1 + x 2 x - 1 = x + 1 + x 2 x - 1 = 2 x + 1 2 x - 1

Answer: x x + 1 + 1 2 x - 1 1 + x = 2 x + 1 2 x - 1 .

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This lesson will cover the basic information about rational expressions and their transformations, as well as examples of the transformation of rational expressions. This topic as if summarizing the topics we have studied so far. Transformations of rational expressions involve addition, subtraction, multiplication, division, raising to the power of algebraic fractions, reduction, factorization, etc. As part of the lesson, we will look at what a rational expression is, and also analyze examples for their transformation.

Topic:Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:Basic information about rational expressions and their transformations

Definition

rational expression is an expression consisting of numbers, variables, arithmetic operations and exponentiation operations.

Consider an example of a rational expression:

Special cases of rational expressions:

1st degree: ;

2. monomial: ;

3. fraction: .

Rational Expression Transformation is a simplification of a rational expression. The order of operations when converting rational expressions: first, there are actions in brackets, then multiplication (division), and then addition (subtraction) operations.

Let's consider some examples on transformation of rational expressions.

Example 1

Solution:

Let's solve this example step by step. The action in parentheses is performed first.

Answer:

Example 2

Solution:

Answer:

Example 3

Solution:

Answer: .

Note: perhaps, at the sight of this example, an idea occurred to you: reduce the fraction before reducing to a common denominator. Indeed, it is absolutely correct: first, it is desirable to simplify the expression as much as possible, and then transform it. Let's try to solve the same example in the second way.

As you can see, the answer turned out to be absolutely similar, but the solution turned out to be somewhat simpler.

In this lesson, we looked at rational expressions and their transformations, as well as several specific examples of these transformations.

Bibliography

1. Bashmakov M.I. Algebra 8th grade. - M.: Enlightenment, 2004.

2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 8. - 5th ed. - M.: Education, 2010.

This lesson will cover the basic information about rational expressions and their transformations, as well as examples of the transformation of rational expressions. This topic summarizes the topics we have studied so far. Transformations of rational expressions involve addition, subtraction, multiplication, division, raising to the power of algebraic fractions, reduction, factorization, etc. As part of the lesson, we will look at what a rational expression is, and also analyze examples for their transformation.

Topic:Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:Basic information about rational expressions and their transformations

Definition

rational expression is an expression consisting of numbers, variables, arithmetic operations and exponentiation.

Consider an example of a rational expression:

Special cases of rational expressions:

1st degree: ;

2. monomial: ;

3. fraction: .

Rational Expression Transformation is a simplification of a rational expression. The order of operations when converting rational expressions: first, there are actions in brackets, then multiplication (division), and then addition (subtraction) operations.

Let's consider some examples on transformation of rational expressions.

Example 1

Solution:

Let's solve this example step by step. The action in parentheses is performed first.

Answer:

Example 2

Solution:

Answer:

Example 3

Solution:

Answer: .

Note: perhaps, at the sight of this example, an idea occurred to you: reduce the fraction before reducing to a common denominator. Indeed, it is absolutely correct: first, it is desirable to simplify the expression as much as possible, and then transform it. Let's try to solve the same example in the second way.

As you can see, the answer turned out to be absolutely similar, but the solution turned out to be somewhat simpler.

In this lesson, we looked at rational expressions and their transformations, as well as several specific examples of these transformations.

Bibliography

1. Bashmakov M.I. Algebra 8th grade. - M.: Enlightenment, 2004.

2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 8. - 5th ed. - M.: Education, 2010.