Converting rational expressions. Transformation of rational expressions, types of transformations, examples How to understand the transformation of rational expressions

The article talks about the transformation of rational expressions. Let's consider the types of rational expressions, their transformations, groupings, and bracketing the common factor. Let's learn to represent fractional rational expressions in the form of rational fractions.

Yandex.RTB R-A-339285-1

Definition and examples of rational expressions

Definition 1

Expressions that are made up of numbers, variables, parentheses, powers with the operations of addition, subtraction, multiplication, division with the presence of a fraction line 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 are not divided into expressions with variables. The study of rational expressions begins in grade 8, where they are called fractional rational expressions. Particular attention is paid to fractions in the numerator, which are transformed using transformation rules.

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

Main types of transformations of rational expressions

Rational expressions are used to perform identical transformations, groupings, bringing similar ones, and performing other operations with numbers. The purpose of such expressions is simplification.

Example 1

Convert the 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 between 3 x x y - 1 and 2 x x y - 1. We notice that their denominator is identical. This means that the reduction of similar terms will take 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

Convert 2 x y 4 (- 4) x 2: (3 x - x) .

Solution

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

We 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 numerical factors with the variable x, after which 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

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

Solution

First, we transform 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 parentheses are done first. In the numerator, operations 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 difference of squares formula 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 .

Rational fraction representation

Algebraic fractions are most often simplified when solved. 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 ultimately give a rational fraction.

Example 4

Present 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 primarily 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 present the obtained result with the original one. 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 which 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, the numerator of which has x x + 1 + 1, and 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 this differently.

Instead of dividing by 2 x - 1 1 + x, we multiply by its inverse 1 + x 2 x - 1. Let us apply the distribution property and find 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 .

If you notice an error in the text, please highlight it and press Ctrl+Enter

Lesson and presentation on the topic: "Transformation of rational expressions. Examples of problem solving"

Additional materials
Dear users, do not forget to leave your comments, reviews, wishes. All materials have been checked by an anti-virus program.

Educational aids and simulators in the Integral online store for grade 8
Manual for the textbook Muravin G.K. A manual for the textbook by Makarychev Yu.N.

The concept of rational expression

The concept of "rational expression" is similar to the concept of "rational fraction". The expression is also represented as a fraction. Only our numerators are not numbers, but various kinds of expressions. Most often these are polynomials. An algebraic fraction is a fractional expression consisting of numbers and variables.

When solving many problems in the elementary grades, after performing arithmetic operations, we received specific numerical values, most often fractions. Now after performing the operations we will obtain algebraic fractions. Guys, remember: to get the correct answer, you need to simplify the expression you are working with as much as possible. One must obtain the smallest degree possible; identical expressions in numerators and denominators should be reduced; with expressions that can be collapsed, you must do so. That is, after performing a series of actions, we should obtain the simplest possible algebraic fraction.

Procedure with rational expressions

The procedure for performing operations with rational expressions is the same as for arithmetic operations. First, the operations in parentheses are performed, then multiplication and division, exponentiation, and finally addition and subtraction.

To prove an identity means to show that for all values ​​of the variables the right and left sides are equal. There are a lot of examples of proving identities.

The main ways to solve identities include.

• Transform the left side to be equal to the right side.
• Transform the right side to be equal to the left.
• Transform the left and right sides separately until you get the same expression.
• The right side is subtracted from the left side, and the result should be zero.

Converting rational expressions. Examples of problem solving

Example 1.
Prove the identity:

$(\frac(a+5)(5a-1)+\frac(a+5)(a+1)):(\frac(a^2+5a)(1-5a))+\frac(a ^2+5)(a+1)=a-1$.

Solution.
Obviously, we need to transform the left side.
First, let's do the steps in parentheses:

1) $\frac(a+5)(5a-1)+\frac(a+5)(a+1)=\frac((a+5)(a+1)+(a+5)(5a -1))((a+1)(5a-1))=$
$=\frac((a+5)(a+1+5a-1))((a+1)(5a-1))=\frac((a+5)(6a))((a+1 )(5a-1))$

.

You should try to apply common factors to the maximum.
2) Transform the expression by which we divide:

$\frac(a^2+5a)(1-5a)=\frac(a(a+5))((1-5a)=\frac(a(a+5))(-(5a-1) )$

.
3) Perform the division operation:

$\frac((a+5)(6a))((a+1)(5a-1)):\frac(a(a+5))(-(5a-1))=\frac((a +5)(6a))((a+1)(5a-1))*\frac(-(5a-1))(a(a+5))=\frac(-6)(a+1) $.

4) Perform the addition operation:

$\frac(-6)(a+1)+\frac(a^2+5)(a+1)=\frac(a^2-1)(a+1)=\frac((a-1 )(a+1))(a+))=a-1$.

The right and left parts coincided. This means the identity is proven.
Guys, when solving this example we needed knowledge of many formulas and operations. We see that after the transformation, the large expression has turned into a very small one. When solving almost all problems, transformations usually lead to simple expressions.

Example 2.
Simplify the expression:

$(\frac(a^2)(a+b)-\frac(a^3)(a^2+2ab+b^2)):(\frac(a)(a+b)-\frac( a^2)(a^2-b^2))$.

Solution.
Let's start with the first brackets.

1. $\frac(a^2)(a+b)-\frac(a^3)(a^2+2ab+b^2)=\frac(a^2)(a+b)-\frac (a^3)((a+b)^2)=\frac(a^2(a+b)-a^3)((a+b)^2)=$
$=\frac(a^3+a^2 b-a^3)((a+b)^2)=\frac(a^2b)((a+b)^2)$.

2. Transform the second brackets.

$\frac(a)(a+b)-\frac(a^2)(a^2-b^2)=\frac(a)(a+b)-\frac(a^2)((a-b )(a+b))=\frac(a(a-b)-a^2)((a-b)(a+b))=$
$=\frac(a^2-ab-a^2)((a-b)(a+b))=\frac(-ab)((a-b)(a+b))$.

3. Let's do the division.

$\frac(a^2b)((a+b)^2):\frac(-ab)((a-b)(a+b))=\frac(a^2b)((a+b)^2 )*\frac((a-b)(a+b))((-ab))=$
$=-\frac(a(a-b))(a+b)$

.

Answer: $-\frac(a(a-b))(a+b)$.

Example 3.
Follow these steps:

$\frac(k-4)(k-2):(\frac(80k)((k^3-8)+\frac(2k)(k^2+2k+4)-\frac(k-16 )(2-k))-\frac(6k+4)((4-k)^2)$.

Solution.
As always, you need to start with the brackets.

1. $\frac(80k)(k^3-8)+\frac(2k)(k^2+2k+4)-\frac(k-16)(2-k)=\frac(80k)( (k-2)(k^2+2k+4)) +\frac(2k)(k^2+2k+4)+\frac(k-16)(k-2)=$

$=\frac(80k+2k(k-2)+(k-16)(k^2+2k+4))((k-2)(k^2+2k+4))=\frac(80k +2k^2-4k+k^3+2k^2+4k-16k^2-32k-64)((k-2)(k^2+2k+4))=$

$=\frac(k^3-12k^2+48k-64)((k-2)(k^2+2k+4))=\frac((k-4)^3)((k-2 )(k^2+2k+4))$.

2. Now let's do the division.

$\frac(k-4)(k-2):\frac((k-4)^3)((k-2)(k^2+2k+4))=\frac(k-4)( k-2)*\frac((k-2)(k^2+2k+4))((k-4)^3)=\frac((k^2+2k+4))((k- 4)^2)$.

3. Let's use the property: $(4-k)^2=(k-4)^2$.
4. Let's perform the subtraction operation.

$\frac((k^2+2k+4))((k-4)^2)-\frac(6k+4)((k-4)^2)=\frac(k^2-4k) ((k-4)^2)=\frac(k(k-4))((k-4)^2)=\frac(k)(k-4)$.

As we said earlier, you need to simplify the fraction as much as possible.
Answer: $\frac(k)(k-4)$.

Problems to solve independently

1. Prove the identity:

$\frac(b^2-14)(b-4)-(\frac(3-b)(7b-4)+\frac(b-3)(b-4))*\frac(4-7b )(9b-3b^2)=b+4$.

2. Simplify the expression:

$\frac(4(z+4)^2)(z-2)*(\frac(z)(2z-4)-\frac(z^2+4)(2z^2-8)-\frac (2)(z^2+2z))$.

3. Follow these steps:

$(\frac(a-b)(a^2+2ab+b^2)-\frac(2a)((a-b)(a+b))+\frac(a-b)((a-b)^2))*\ frac(a^4-b^4)(8ab^2)+\frac(2b^2)(a^2-b^2)$.

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

Subject: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 the operation of exponentiation.

Let's look at an example of a rational expression:

Special cases of rational expressions:

1st degree: ;

2. monomial: ;

3. fraction: .

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

Let's look at several examples of transforming rational expressions.

Example 1

Solution:

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

Answer:

Example 2

Solution:

Answer:

Example 3

Solution:

Answer: .

Note: Perhaps, when you saw this example, an idea arose: reduce the fraction before reducing it to a common denominator. Indeed, it is absolutely correct: first it is advisable to simplify the expression as much as possible, and then transform it. Let's try to solve this 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.: Education, 2004.

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

This article is dedicated to transformation of rational expressions, mostly fractionally rational, is one of the key issues in the 8th grade algebra course. First, we recall what type of expressions are called rational. Next we will focus on carrying out standard transformations with rational expressions, such as grouping terms, putting common factors out of brackets, bringing similar terms, etc. Finally, we will learn to represent fractional rational expressions as rational fractions.

Page navigation.

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 composed of numbers, variables, parentheses, powers with integer exponents, connected using arithmetic signs +, −, · and:, where division can be indicated by a fraction line, are called rational expressions.

Here are some examples of rational expressions: .

Rational expressions begin to be studied purposefully in the 7th grade. Moreover, in the 7th grade one learns 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 sequentially studied, as well as the principles of performing actions with them. All this knowledge ultimately allows you to perform transformations of entire expressions.

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

The acquired skills allow you to move on to transforming rational expressions of any 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 whole expressions and algebraic fractions.

Main types of transformations of rational expressions

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

Example.

.

Solution.

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

Answer:

.

It is clear that when carrying out transformations with rational expressions, as well as with any other expressions, you need to remain within the accepted order of performing actions.

Example.

Perform a rational expression transformation.

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 obtained result into 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 property of division by a product: .

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

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

Answer:

Example.

Convert rational expression .

Solution.

First we transform the numerator and denominator. This order of transformation of fractions is explained by the fact that the line of a fraction is essentially another designation for division, and the original rational expression is essentially a quotient of the form , and the actions in parentheses are performed 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 in the form of a product: suddenly it is possible to reduce an algebraic fraction. To do this, we will use in the numerator difference of squares formula, and in the denominator we take the two out of brackets, we have .

Answer:

.

So, the initial acquaintance with the transformation of rational expressions can be considered completed. Let's move on, so to speak, to the sweetest part.

Rational fraction representation

Most often, the ultimate goal of transforming expressions is to simplify their appearance. In this light, the simplest form to which a fractional rational expression can be converted is a rational (algebraic) fraction, and in the particular case a polynomial, monomial or number.

Is it possible to represent any rational expression as a rational fraction? The answer is yes. Let us explain why this is so.

As we have already said, every rational expression can be considered as polynomials and rational fractions connected by plus, minus, multiply and divide signs. All corresponding operations with polynomials yield a polynomial or rational fraction. In turn, any polynomial can be converted into an algebraic fraction by writing it with the denominator 1. And adding, subtracting, multiplying and dividing rational fractions results 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 the product of fractions of the form . According to the order of operations, we must first perform multiplication, and only then addition.

We start with multiplying algebraic fractions:

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

We came to the subtraction of algebraic fractions with different denominators:

So, having performed operations with rational fractions that make up the original rational expression, we presented it in the form of a rational fraction.

Answer:

.

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

Example.

Express a rational expression as a rational fraction.