Explanation of the topic of transformation of rational expressions. Transformation of rational expressions, types of transformations, examples

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.

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Definition and examples of rational expressions

Definition 1

Expressions that are composed 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 .

Let us 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 .

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

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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:

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

From the school algebra course we move on to specifics. In this article we will study in detail a special type of rational expressions - rational fractions, and also consider what characteristic identical rational fraction conversions take place.

Let us immediately note that rational fractions in the sense in which we define them below are called algebraic fractions in some algebra textbooks. That is, in this article we will understand rational and algebraic fractions as the same thing.

As usual, let's start with a definition and examples. Next we’ll talk about bringing a rational fraction to a new denominator and changing the signs of the members of the fraction. After this, we will look at how to reduce fractions. Finally, let's look at representing a rational fraction as a sum of several fractions. We will provide all information with examples and detailed descriptions of solutions.

Page navigation.

Definition and examples of rational fractions

Rational fractions are studied in 8th grade algebra lessons. We will use the definition of a rational fraction, which is given in the algebra textbook for 8th grade by Yu. N. Makarychev et al.

This definition does not specify whether the polynomials in the numerator and denominator of a rational fraction must be polynomials of the standard form or not. Therefore, we will assume that the notations for rational fractions can contain both standard and non-standard polynomials.

Here are a few examples of rational fractions. So, x/8 and - rational fractions. And fractions and do not fit the stated definition of a rational fraction, since in the first of them the numerator does not contain a polynomial, and in the second, both the numerator and the denominator contain expressions that are not polynomials.

Converting the numerator and denominator of a rational fraction

The numerator and denominator of any fraction are self-sufficient mathematical expressions; in the case of rational fractions, these are polynomials; in a particular case, monomials and numbers. Therefore, identical transformations can be carried out with the numerator and denominator of a rational fraction, as with any expression. In other words, the expression in the numerator of a rational fraction can be replaced by an identically equal expression, just like the denominator.

You can perform identical transformations in the numerator and denominator of a rational fraction. For example, in the numerator you can group and reduce similar terms, and in the denominator you can replace the product of several numbers with its value. And since the numerator and denominator of a rational fraction are polynomials, it is possible to perform transformations characteristic of polynomials with them, for example, reduction to a standard form or representation in the form of a product.

For clarity, let's consider solutions to several examples.

Example.

Convert rational fraction so that the numerator contains a polynomial of standard form, and the denominator contains the product of polynomials.

Solution.

Reducing rational fractions to a new denominator is primarily used in adding and subtracting rational fractions.

Changing signs in front of a fraction, as well as in its numerator and denominator

The main property of a fraction can be used to change the signs of the members of a fraction. Indeed, multiplying the numerator and denominator of a rational fraction by -1 is equivalent to changing their signs, and the result is a fraction identically equal to the given one. This transformation has to be used quite often when working with rational fractions.

Thus, if you simultaneously change the signs of the numerator and denominator of a fraction, you will get a fraction equal to the original one. This statement is answered by equality.

Let's give an example. A rational fraction can be replaced by an identically equal fraction with changed signs of the numerator and denominator of the form.

With fractions, you can carry out another identical transformation, in which the sign of either the numerator or the denominator changes. Let us state the corresponding rule. If you replace the sign of a fraction together with the sign of the numerator or denominator, you get a fraction that is identically equal to the original one. The written statement corresponds to the equalities and .

Proving these equalities is not difficult. The proof is based on the properties of multiplication of numbers. Let's prove the first of them: . Using similar transformations, the equality is proved.

For example, a fraction can be replaced by the expression or.

To conclude this point, we present two more useful equalities and . That is, if you change the sign of only the numerator or only the denominator, the fraction will change its sign. For example, And .

The considered transformations, which allow changing the sign of the terms of a fraction, are often used when transforming fractional rational expressions.

Reducing rational fractions

The following transformation of rational fractions, called reduction of rational fractions, is based on the same basic property of a fraction. This transformation corresponds to the equality , where a, b and c are some polynomials, and b and c are non-zero.

From the above equality it becomes clear that reducing a rational fraction implies getting rid of the common factor in its numerator and denominator.

Example.

Cancel a rational fraction.

Solution.

The common factor 2 is immediately visible, let’s perform a reduction by it (when writing, it is convenient to cross out the common factors that are being reduced by). We have . Since x 2 =x x and y 7 =y 3 y 4 (see if necessary), it is clear that x is a common factor of the numerator and denominator of the resulting fraction, as is y 3. Let's reduce by these factors: . This completes the reduction.

Above we carried out the reduction of rational fractions sequentially. Or it was possible to perform the reduction in one step, immediately reducing the fraction by 2 x y 3. In this case, the solution would look like this: .

Answer:

.

When reducing rational fractions, the main problem is that the common factor of the numerator and denominator is not always visible. Moreover, it does not always exist. In order to find a common factor or verify its absence, you need to factor the numerator and denominator of a rational fraction. If there is no common factor, then the original rational fraction does not need to be reduced, otherwise, reduction is carried out.

Various nuances can arise in the process of reducing rational fractions. The main subtleties are discussed in the article reducing algebraic fractions using examples and in detail.

Concluding the conversation about the reduction of rational fractions, we note that this transformation is identical, and the main difficulty in its implementation lies in factoring the polynomials in the numerator and denominator.

Representation of a rational fraction as a sum of fractions

Quite specific, but in some cases very useful, is the transformation of a rational fraction, which consists in its representation as the sum of several fractions, or the sum of an entire expression and a fraction.

A rational fraction, the numerator of which contains a polynomial representing the sum of several monomials, can always be written as a sum of fractions with the same denominators, the numerators of which contain the corresponding monomials. For example, . This representation is explained by the rule for adding and subtracting algebraic fractions with like denominators.

In general, any rational fraction can be expressed as a sum of fractions in many different ways. For example, the fraction a/b can be represented as the sum of two fractions - an arbitrary fraction c/d and a fraction equal to the difference between the fractions a/b and c/d. This statement is true, since the equality holds . For example, a rational fraction can be represented as a sum of fractions in various ways: Let's imagine the original fraction as the sum of an integer expression and a fraction. By dividing the numerator by the denominator with a column, we get the equality . The value of the expression n 3 +4 for any integer n is an integer. And the value of a fraction is an integer if and only if its denominator is 1, −1, 3, or −3. These values ​​correspond to the values ​​n=3, n=1, n=5 and n=−1, respectively.

Answer:

−1 , 1 , 3 , 5 .

References.

  • 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. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 13th ed., rev. - M.: Mnemosyne, 2009. - 160 pp.: ill. ISBN 978-5-346-01198-9.
  • Mordkovich A. G. Algebra. 8th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemosyne, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
  • Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

ALGEBRA
All lessons for 8th grade

Lesson #19

Subject. Identical transformations of rational expressions

Goal: to consolidate students’ knowledge about algorithms for identical transformations of rational expressions, methods for transforming the ratio of two fractional expressions and schemes for applying the properties of arithmetic operations when transforming rational expressions.

Lesson type: knowledge correction, skill development.

Visualization and equipment: basic summary “Identical transformations of algebraic expressions.”

Lesson progress

I. Organizational stage

II. Checking homework

Exercises on the use of techniques for transforming expressions that have the form of a ratio of two rational expressions (“four-story fractions”) are subject to careful analysis. To make this work more conscious, you can ask students to fill out the table:

It is clear that this work can only be effective in the case of further correction.

For students who have well mastered the techniques of working with expressions that are subject to control at this stage of the lesson, the teacher can offer additional tasks of this type and evaluate their implementation.

III. Formulation of revenge and lesson objectives

A check of homework completion and an analysis of possible errors in themselves create motivation for students to work to eliminate the cause of errors (knowledge correction), as well as improve skills (skill formation). Achieving the best results of this activity - correcting knowledge and developing students’ skills to transform rational expressions using the studied algorithms for performing arithmetic operations with rational fractions - constitutes the main didactic purpose of the lesson.

IV. Updating basic knowledge and skills

@ In order for students to successfully perceive the educational material, such knowledge should be activated before studying the lesson material. and student skills: rules for performing arithmetic operations with rational numbers and the order of performing actions in numerical expressions containing actions of varying degrees; identical transformations of entire expressions; converting the sum, difference, product and fraction of two rational fractions into a rational fraction, as well as converting a rational fraction using the basic property of a rational fraction (raising a rational fraction to a new denominator, raising several rational fractions to a new least common denominator).

Taking into account the didactic goal (emphasis on correctional work) and in order to diversify the forms of work in the lesson, you can invite students at this stage of the lesson to conduct a quick survey (or conduct an interactive exercise “Microphone”); the main condition is a clear and concise answer to the question.

1. How is the main property of a fraction formulated?

2. What happens to the sign of a fraction if you replace the sign of its numerator; denominator; numerator and denominator?

3. How do I add fractions with like denominators?

4. How to subtract fractions with like denominators?

5. How to add fractions with different denominators? Give an example of the fractions: a) and ; b) and .

6. How to multiply two fractions?

7. What rule do you know for raising a fraction to a power?

8. Formulate the rule for dividing fractions.

9. Tell us about the procedure for converting expressions: a) ; b) ; V) .

V. Formation of skills

Performing oral exercises

1. Write the following expression as an irreducible fraction:

A) ; b) ; V) ; G) ; d) ; e) ; and) ; h) ; And) ; To) ; l) .

2. Name the lowest common denominator of fractions (expressions):

a) and ; b) a; And ; c) and ; And ; d) and .

3. At what values ​​of the variable does the value of the fraction equal zero?

Doing writing exercises

During a lesson in correcting knowledge and practicing skills, it would be logical to ask students to solve exercises with approximately the following content:

1. Converting a rational expression to a rational fraction (according to the general scheme drawn up in lesson 17).

1) Simplify the expression: a) ; b) ; V) .

2) Simplify the expression: a) ; b) ; V) .

3) Simplify the expression:
A) ; b) ; V) ; G) .

4) Follow these steps:
A) ; b) ; V) ; G) .

5) Simplify the expression:
A) ; b) ; V) ;

G) ; d) ; e) .

2. Representation of the ratio of fractional rational expressions in the form of a ratio of polynomials (using the basic property of a fraction).

1) Present as a rational fraction: .

2) Find the meaning of the expression:

a) with a = , b = ; b) at a = -8, b = 0.6.

3) Present as a rational fraction:

A) ; b) ; V) ; G) .

3. Evidence that the value of the expression does not depend on the value of the variable.

1) Prove that for all possible values ​​of the letters, the value of the expression equals 0.

2) Prove that for any natural number n the value of the expression is a natural number.

4. Proofs of identities.
Prove the identity:

A) ;

b) .

5. Repetition exercises (especially on finding the ODZ of a rational expression and finding the values ​​of variables at which the value of the expression is zero).

6. Logical exercises and tasks of an increased level of complexity for students with sufficient and high levels of knowledge.

1) Present the expression as a rational fraction: a) ; b) .

2) Prove that for all admissible values ​​of the variables, the value of the expression does not depend on a and b.

3) is the expression missing?

@ As mentioned above, the task of converting rational expressions into rational fractions in the general case is a rather complex task, since it requires fluent mastery of algorithms for performing various arithmetic operations with rational fractions, as well as a fairly high level of ability to apply these algorithms in practice and switch from one algorithm to another. Therefore, the teacher chooses the level of difficulty of tasks depending on the level of knowledge and skills of the students, without underestimating the requirements for students, but at the same time creating a situation of success. In order to prepare students for the perception of the next section (“Rational Equations”), they should continue to solve exercises on finding the ODZ of a rational expression and finding the values ​​of variables at which the value of the expression is equal to zero.

3. Repeat: definition of a rational whole, rational and fractional rational expressions, ODZ of a rational expression; definition of an equation, equivalence properties of equations, the concept of a linear equation with one variable and an algorithm for solving a linear equation; solve linear equations (including equations with parameters); repeat the content of the concept of “proportion” and the main property of proportion, solve several equations for the application of this property (see grade 6).


In the previous lesson, the concept of a rational expression was already introduced; in today's lesson we continue to work with rational expressions and focus on their transformations. Using specific examples, we will consider methods for solving problems involving transformations of rational expressions and proving the identities associated with them.

Subject:Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:Converting rational expressions

Let us first recall the definition of a rational expression.

Definition.Rationalexpression- an algebraic expression that does not contain roots and includes only the operations of addition, subtraction, multiplication and division (raising to a power).

By the concept of “transforming a rational expression” we mean, first of all, its simplification. And this is carried out in the order of actions known to us: first the actions in brackets, then product of numbers(exponentiation), dividing numbers, and then adding/subtracting operations.

The main goal of today's lesson will be to gain experience in solving more complex problems of simplifying rational expressions.

Example 1.

Solution. At first it may seem that these fractions can be reduced, since the expressions in the numerators of fractions are very similar to the formulas for the perfect squares of their corresponding denominators. In this case, it is important not to rush, but to separately check whether this is so.

Let's check the numerator of the first fraction: . Now the second numerator: .

As you can see, our expectations were not met, and the expressions in the numerators are not perfect squares, since they do not have doubling of the product. Such expressions, if you remember the 7th grade course, are called incomplete squares. You should be very careful in such cases, because confusing the formula of a complete square with an incomplete one is a very common mistake, and such examples test the student’s attentiveness.

Since reduction is impossible, we will perform the addition of fractions. The denominators do not have common factors, so they are simply multiplied to obtain the lowest common denominator, and the additional factor for each fraction is the denominator of the other fraction.

Of course, you can then open the brackets and then bring similar terms, however, in this case you can get by with less effort and notice that in the numerator the first term is the formula for the sum of cubes, and the second is the difference of cubes. For convenience, let us recall these formulas in general form:

In our case, the expressions in the numerator are collapsed as follows:

, the second expression is similar. We have:

Answer..

Example 2. Simplify rational expression .

Solution. This example is similar to the previous one, but here it is immediately clear that the numerators of the fractions contain partial squares, so reduction at the initial stage of the solution is impossible. Similarly to the previous example, we add the fractions:

Here, similarly to the method indicated above, we noticed and collapsed the expressions using the formulas for the sum and difference of cubes.

Answer..

Example 3. Simplify a rational expression.

Solution. You can notice that the denominator of the second fraction is factorized using the sum of cubes formula. As we already know, factoring denominators is useful for further finding the lowest common denominator of fractions.

Let us indicate the lowest common denominator of the fractions, it is equal to: , since it is divided by the denominator of the third fraction, and the first expression is generally an integer, and any denominator is suitable for it. Having indicated the obvious additional factors, we write:

Answer.

Let's consider a more complex example with “multi-story” fractions.

Example 4. Prove the identity for all permissible values ​​of the variable.

Proof. To prove this identity, we will try to simplify its left side (complex) to the simple form that is required of us. To do this, we will perform all the operations with fractions in the numerator and denominator, and then divide the fractions and simplify the result.

Proven for all permissible values ​​of the variable.

Proven.

In the next lesson we will look in detail at more complex examples of converting rational expressions.

References

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.

3. Nikolsky S.M., Potapov M.A., Reshetnikov N.N., Shevkin A.V. Algebra 8th grade. Textbook for general education institutions. - M.: Education, 2006.

2. Lesson developments, presentations, lesson notes ().

Homework

1. No. 96-101. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. Algebra 8. - 5th ed. - M.: Education, 2010.

2. Simplify the expression .

3. Simplify the expression.

4. Prove the identity.



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