Permissible values \u200b\u200bof the variable for algebraic fraction. Multiplication, division and reduction of algebraic fractions

In § 42, it was said that if the division of polynomials cannot be performed aimed, the private is written in the form of a fractional expression in which divisible is a numerator, and the divisor is denominator.

Examples of fractional expressions:

The numerator and denominator of the fractional expression and themselves can be fractional expressions, for example:

Of fractional algebraic expressions, most often have to deal with those in which the numerator and the denominator are polynomials (in particular, and single-pans). Each such expression is called an algebraic fraction.

Definition. An algebraic expression, which is a fraction, the numerator and denominator of which are polynomials, is called an algebraic fraction.

As in arithmetic, the numerator and denominator of the algebraic fraction are called the members of the fraction.

In the future, having studied the actions on algebraic fractions, we will be able to transform any fractional expression using identical transformations into an algebraic fraction.

Examples of algebraic fractions:

Note that a whole expression, that is, a polynomial can be written in the form of a fraction, for this it is enough to write in the numerator this expression, and in the denominator 1. For example:

2. Permissible values \u200b\u200bof letters.

The letters included in the numerator can take any values \u200b\u200b(if any additional limitations are not entered).

For the letters included in the denominator, only those values \u200b\u200bthat are not paid to zero are valid. Therefore, in the future, we will always assume that the denominator of the algebraic fraction is not equal to zero.

Topic: Repetition of the 8th grade algebra

Lesson: Algebraic fractions

To begin with, let's remember what is algebraic fractions. Algebraic fraction call the expression of the view where - polynomials - Numerator, - denominator.

Since - polynomials, it is necessary to keep in mind the standard actions possible with polynomials, namely: bringing to standard form, expansion of multipliers, as well as a reduction in the number and denominator.

Example №1

Reduce fraction

We use the formulas of abbreviated multiplication for the square of the sum and the difference of squares.

Comments: Initially, we launched a fraction on the factors using the formulas of abbreviated multiplication, and then we used one of the main properties of the fraction: and the numerator, and the denominator of the algebraic fraction can be multiplied or divided into one and the same polynomial, including the number that is not equal to 0 Thus, it turns out that we are a numerator, and the denominator was divided into a polynomial, so it is necessary to consider that this polynomial is not equal to 0, that is,.

Example number 2.

From the condition, we are not yet clear to us, what is the connection between these two functions. To do this, we need to simplify the first of them by expanding the factors.

however, it is necessary not to forget about the cutting condition of the fraction, that is, about the fact that

After all the abbreviations, we get that

only with the difference that .

Build a graph of two functions.

We see a bright difference between these two charts: in fact, they are the same, but in the first chart we need to buy a point with the coordinate (1; 0), since this accurate is not included in the OTZ of the first function.

Total, we looked at what fraction was, we decided a couple of examples about how important to follow the area of \u200b\u200bdefinition (the area of \u200b\u200bpermissible values), i.e., for those values \u200b\u200bthat can be taken.

Now let's turn to the question of what actions can be made with algebraic dolls, in addition to those already mentioned above.

Naturally, algebraic fractions, as well as arithmetic fractions, can be added, deduct, multiply, divide, to be deducted, while obtaining rational algebraic expressions (such expressions that are composed of among the numbers, variables using arithmetic operations and erection in natural degree ). After certain simplifications, such expressions are reduced to fractions for which the initial expressions are also algebraic fractions.

The list of actions / conditions with which you can encounter solving tasks for algebraic fractions:

Simplify rational expressions

Prove identity

Solve a rational equation

Simplify / calculate fraction

Example number 3.

Solve the simplest rational equation

The fraction is 0 if and only if the numerator is 0, and the denominator is not equal to 0. In our case, the denominator is equal. It means that the fraction solution is reduced to the linear equation

Example number 4.

Solve equation

First, try to reduce the fraction

Provided that .

Since we have already simplified the fraction on the left side of the original equation, we can substitute a new value and solve the equation.

Now let's try to highlight the full square of the resulting square equation

We use the formula of abbreviated multiplication for square differences

The product is 0 if and only if at least one of the multipliers is 0. In addition, we do not forget that at the beginning we have a condition for the existence of our expression in the form. Write the same system of equations.

\u003d\u003e \u003d\u003e We see that it contradicts our condition that, so we have only one answer.

So, let's look at the features that we have solved above example:

1. The numerator with the difference of cubes and the denominator is desirable to reduce immediately, since this is possible in this case and will greatly simplify the further solution of the equation, but it is necessary to remember that the denomote denominator cannot be equal to 0 and write this condition.

2. Leaving the fraction to the square equation, we remembered one of the solutions square equations - Method of allocation of a full square.

We are with you this lesson They remembered that such an algebraic fraction, which actions need to be produced with a numerator and denominator in solving such fractions, which actions in general can be made with fractions of this species and solved several simple tasks.

Bibliography

1. Bashmakov M.I. Algebra Grade 8. - M.: Enlightenment, 2004.
2. Dorofeyev G.V., Suvorova S.B., Baynovich E.A. and others. Algebra 8. 5 edition. - M.: Enlightenment, 2010.
3. Nikolsky S.M., Potapov MA, Reshetnikov N.N., Shevkin A.V. Algebra Grade 8. Tutorial for general educational institutions. - M.: Education, 2006.

1. All elementary mathematics ().
2. School assistant ().
3. Internet portal testmath.com.ua ().

Homework

From the course algebra school program Go to specific. In this article, we will examine the special type of rational expressions in detail - rational fractionsand also we will analyze what characteristic identical transformation of rational fractions take place.

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

Let us begin with definition and examples. Before talk about bringing rational fraci To the new denominator and the change of signs in the members of the fraction. After that, we will analyze how the frains are reduced. Finally, we will focus on the representation of a rational fraction in the form of a sum of several fractions. All the information will be supplied with examples with detailed descriptions solutions.

Navigating page.

Definition and examples of rational fractions

Rational frarators are studied in the lessons of algebra in grade 8. We will use the definition of rational fraction, which is given in the textbook of algebra for 8 classes Yu. N. Makarychev, etc.

IN this definition It is not specified whether the polynomials in the numerator and the denominator of the rational fraction should be polynomials of the standard type or not. Therefore, we assume that in the records of rational fractions can be found both polynomials of the standard species and not standard.

We give a few examples of rational fractions. So, x / 8 and - rational fractions. And the fraci And they are not suitable for the voiced definition of rational fraction, since in the first of them in the numerator it is not a polynomial, but in the second and in the numerator and in the denominator are expressions that are not polynomials.

Transformation of the numerator and denominator of rational fraction

The numerator and denominator of any fraction are self-sufficient mathematical expressions, in the case of rational fractions, these are polynomials, in the particular case - are unoccupied and numbers. Therefore, with a numerator and denominator of rational fraction, as with any expression, can be carried out identical conversions. In other words, the expression in the rational fraction numerator can be replaced by an identical expression equal to it, as well as the denominator.

In the numerator and denominator of rational fraction, identical conversions can be performed. For example, in the numerator you can carry out a grouping and bringing similar terms, and in the denominator - the product of several numbers replace it with a value. And since the numerator and denominator of rational fraction are polynomials, then with them you can also perform and characteristic of the polynomials of the transformation, for example, bringing to a standard form or representation in the form of a piece.

For clarity, consider solutions to several examples.

Example.

Convert rational fraction So that the polynomial is a polynomial of a standard species in the numerator, and in the denominator - the product of polynomials.

Decision.

The creation of rational fractions to a new denominator is mainly used when adding and subtracting rational fractions.

Changing signs before fraction, as well as in its numeric and denominator

The main property of the fraction can be used to change the signs from the members of the fraction. Indeed, the multiplication of the numerator and the denominator of the rational fraction on -1 is equivalent to the change of their signs, and the result is a fraction, identically equal to this. It is often necessary to contact this transformation when working with rational fractions.

Thus, if you simultaneously change the signs in the numerator and denominator of the fraction, it will turn out the fraction equal to the original one. Equality is responsible for this statement.

Let us give an example. The rational fraction can be replaced by identically equal to the fraction with the changed signs of the numerator and the denominator of the species.

With fractions, one more identical conversion can be carried out at which the sign changes either in the numerator or in the denominator. Let's voice the appropriate rule. If you replace the fraction sign together with the number of the number or denominator, it will turn out to the fraction, identically equal to the source. Recorded statement correspond to equality and.

Proving these equality is not difficult. The proof is based on the multiplication properties of numbers. We prove the first of them :. With the help of similar transformations, equality is proved.

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

In conclusion of this paragraph, we give two more useful equality and. That is, if you change the sign only in the numerator or only by the denominator, the fraction will change its sign. For example, and .

Considered transformations that allow you to change the sign in the members of the fraction, often apply when converting fractional rational expressions.

Reducing rational fractions

At the heart of the following transformation of rational fractions having a name reduction of rational fractions, is also the main property of the fraction. This transformation corresponds to equality where A, B and C are some polynomials, and B and C - nonzero.

From the given equality it becomes clear that the reduction of the rational fraction involves the disposal of the total factor in its numerator and the denominator.

Example.

Reduce the rational fraction.

Decision.

A general multiplier 2 is visible, we will perform a reduction on it (when recording, general factors that are reduced, convenient to cross out). Have . Since x 2 \u003d x · x and y 7 \u003d y 3 · y 4 (see if necessary), it is clear that X is a common multiplier of the numerator and denominator of the resulting fraction, like Y 3. We will reduce these factors: . This reduced reduction.

Above, we have reduced the rational fraction consistently. And it was possible to reduce the reduction in one step, immediately reducing the fraction by 2 · x · y 3. In this case, the solution would look like this: .

Answer:

.

With a reduction in rational fractions, the main problem is that the total multiplier of the numerator and the denominator is not always visible. Moreover, it does not always exist. In order to find a common factor or make sure that it is not necessary for a numerator and denominator of rational fraction to decompose on multipliers. If there is no common factor, then the initial rational fraction does not need a reduction, otherwise there is a reduction.

In the process of reduction of rational fractions, various nuances may occur. The main subtleties on the examples and in the details disassembled in the article reducing algebraic fractions.

Completing the conversation about the reduction of rational fractions, we note that this transformation is identical, and the main complexity in its conduct is to decompose the polynomials in the numerator and denominator.

Representation of rational fraction in the form of the amount of fractions

Quite specific, but in some cases very useful, it turns out to transform a rational fraction, which consists in its representation as a sum of several fractions, or the sum of the whole expression and fraction.

The rational fraction, in the numerator of which there is a polynomial, which is a sum of several universions, can always be written as the amount of fractions with the same denominators, in whose numerators are appropriate. For example, . Such a submission is explained by the rule of addition and subtracting algebraic fractions with the same denominators.

In general, any rational fraction can be represented as a fraction by a variety of different ways. For example, the fraction A / B can be represented as the sum of two fractions - arbitrary fraction C / D and fraction equal to the difference of fractions A / B and C / D. This statement is fair, as there is equality . For example, a rational fraction can be represented as a sum of fractions different ways: Imagine the initial fraction in the form of the sum of the whole expression and the fraction. After dividing the numerator to the denominator, we will get equality . The value of the expression N 3 +4 for any whole n is an integer. And the fraction value is an integer then and only if its denominator is 1, -1, 3 or -3. These values \u200b\u200bcorrespond to n \u003d 3, n \u003d 1, n \u003d 5 and n \u003d -1, respectively.

Answer:

−1 , 1 , 3 , 5 .

Bibliography.

• Algebra: studies. For 8 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov]; Ed. S. A. Telikovsky. - 16th ed. - M.: Enlightenment, 2008. - 271 p. : IL. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. 7th grade. In 2 tsp. 1. Tutorial for students of general educational institutions / A. Mordkovich. - 13th ed., Act. - M.: Mnemozina, 2009. - 160 p.: Il. ISBN 978-5-346-01198-9.
• Mordkovich A. G. Algebra. 8th grade. In 2 tsp. 1. Tutorial for students of general educational institutions / A. Mordkovich. - 11th ed., Ched. - M.: Mnemozina, 2009. - 215 p.: Il. ISBN 978-5-346-01155-2.
• Gusev V. A., Mordkovich A. G. Mathematics (benefit for applicants in technical schools): studies. benefit. - m.; Higher. Shk., 1984.-351 p., Il.

After the initial information obtained about the fractions, we turn to the actions with algebraic fractions. With them you can perform any actions until the extermination. When they are fulfilled, we eventually get an algebraic fraction. All items must be disassembled sequentially.

Actions with algebraic fractions are similar to action with ordinary fractions. Therefore, it is worth noting that the rules are coinciding with any actions performed with them.

Addition of algebraic fractions

Addition can be performed in two cases: with the same denominators, if there are different denominators.

If you need to add fractions with the same denominators, you need to add numerators, and the denominator is left unchanged. This rule allows us to take advantage of fractions and polynomials that are in numerators. We get that

a 2 + a · ba · b - 5 + 2 · a · b + 3 a · b - 5 + 2 · b 4 - 4 a · b - 5 \u003d a 2 + a · b + 2 · a · b + 3 + 2 · b 4 - 4 a · b - 5 \u003d a 2 + 3 · a · b - 1 + 2 · b 4 a · b - 5

If there are patterns of fractions with different numerals, then you need to apply a rule: take advantage of bringing to a common denominator, add the fractions obtained.

Example 1.

It is necessary to make the fraction of the fractions x x 2 - 1 and 3 x 2 - x

Decision

We lead to a common denominator of the form x 2 x · x - 1 · x + 1 and 3 · x + 3 x · (x - 1) · (x + 1).

Do the addition and get that

x 2 x · (x - 1) · (x + 1) + 3 · x + 3 x · (x - 1) · (x + 1) \u003d x 2 + 3 · x + 3 x · (x - 1) · (X + 1) \u003d x 2 + 3 · x + 3 x 3 - x

Answer: x 2 + 3 · x + 3 x 3 - x

The article on the addition and subtraction of such fractions has detailed informationwhere each action produced over the fractions is described in detail. When performing addition, the appearance of a reduced fraction is possible.

Subtraction

Subtraction is performed similar to the addition. With the same denominators, the action is performed only in the numerator, the denominator remains unchanged. With different denominators, bringing to a common one. Only after that you can start computing.

Example 2.

We turn to subtract the fractions a + 5 a 2 + 2 and 1 - 2 · a 2 + A A 2 + 2.

Decision

It can be seen that the denominators are identical, which means A + 5 A 2 + 2 - 1 - 2 · A 2 + AA 2 + 2 \u003d A + 5 - (1 - 2 · A 2 + A) A 2 + 2 \u003d 2 · A 2 + 4 A 2 + 2.

We will reduce the fraction 2 · a 2 + 4 A 2 + 2 \u003d 2 · A 2 + 2 A 2 + 2 \u003d 2.

Answer: 2.

Example 3.

Perform subtraction 4 5 · x and 3 x - 1.

Decision

Dangerors are different, so we give a total of 5 · x · (x - 1), we obtain 4 5 · x \u003d 4 · x - 1 5 · x · (x - 1) \u003d 4 · x - 4 5 · x · (x - 1) and 3 x - 1 \u003d 3 · 5 · x (x - 1) · 5 · x \u003d 15 · x 5 · x · (x - 1).

Now performed

4 5 · x - 3 x - 1 \u003d 4 · x - 4 5 · x · (x - 1) - 15 · x 5 · x · (x - 1) \u003d 4 · x - 4 - 15 · x 5 · x · (X - 1) \u003d \u003d - 4 - 11 · x 5 · x · (x - 1) \u003d - 4 - 11 · x 5 · x 2 - 5 · x

Answer: - 4 - 11 · x 5 · x 2 - 5 · x

Detailed information is indicated in the article on the addition and subtraction of algebraic fractions.

Multiplication of algebraic fractions

With fractions, it is possible to multiply with a similar multiplication of ordinary fractions: in order to multiply the fraction, it is necessary to multiply the numerators and denominators separately.

Consider an example of such a plan.

Example 4.

At multiplication 2 x + 2 on x - x · y y y, we obtain that 2 x + 2 · x - x · y y \u003d 2 · (x - x · y) (x + 2) · y.

Now it is necessary to perform transformations, that is, multiplying is unknown to the polynomial. We get that

2 · x - x · y (x + 2) · y \u003d 2 · x - 2 · x · y x · y + 2 · y

It should be pre-decomposition of a fraction on polynomials in order to simplify the fraction. After you can make a reduction. We have that

2 · x 3 - 8 · x 3 · x · y - y · 6 · y 5 x 2 + 2 · x \u003d 2 · x · (x - 2) · (x + 2) y · (3 · x - 1 ) · 6 · y 5 x · (x + 2) \u003d \u003d 2 · x · (x - 2) · (x + 2) · 6 · y 5 y · (3 · x - 1) · x · x + 2 \u003d 12 · (x - 2) · y 4 3 · x - 1 \u003d 12 · x · y 4 - 24 · y 4 3 · x - 1

A detailed consideration of this action can be found in the article by multiplication and division of fractions.

Division

Consider division with algebraic fractions. Apply the rule: in order to divide the fractions, it is necessary to multiply the first to the opposite second.

The fraction that the inverse of this is considered to fraction with the channeled plates with a numerator and denominator. That is, this fraction is called the convergent.

Consider an example.

Example 5.

Perform the division x 2 - x · y 9 · y 2: 2 · x 3 · y.

Decision

Then reverse 2 · x 3 · y The fraction is recorded as 3 · y 2 · x. So we obtain that x 2 - x · y 9 · y 2: 2 · x 3 · y \u003d x 2 - x · y 9 · y 2 · 3 · y 2 · x \u003d x · x - y · 3 · y 9 · Y 2 · 2 · x \u003d x - y 6 · y.

Answer: x 2 - x · y 9 · y 2: 2 · x 3 · y \u003d x - y 6 · y

Construction of algebraic fractions to the degree

If there is a natural degree, then it is necessary to apply a rule of action to the natural degree. With such calculations, we use the rule: when the degree is erected, the numerator and the denominator should be separated separately to the degree, after which write the result.

Example 6.

Consider on the example of the fraction 2 · x x - y. If it is necessary to build it into a degree of equal to 2, then perform actions: 2 · x x - y 2 \u003d 2 · x 2 (x - y) 2. After that, we are erected into a degree obtained unrochene. After performing actions, we obtain that the fraction will take a form 4 · x 2 x 2 - 2 · x · y + y 2.

A detailed solution of such examples is considered in the article on the construction of an algebraic fraction.

When working with a degree of fraction, it is necessary to remember that the numerator and the denominator are separately elevated into the degree. This will noticeably simplifies the process of solving and further simplify the fraction. It is worth paying attention to the sign before the degree. If there is a "minus" sign, then such a fraction should be turned over for ease of calculation.

If you notice a mistake in the text, please select it and press Ctrl + Enter

When the student goes into senior school, mathematics is divided into 2 subjects: algebra and geometry. Concepts are becoming more and more tasks. In some, there are difficulties with the perception of fractions. They missed the first lesson on this topic, and voila. Fruit? The question that will torment throughout the whole school life.

The concept of algebraic fraci

Let's start with the definition. Under algebraic fractionit is understood as the expression P / Q, where p is a numerator, and the q - denominator. Under an alphabone record, a number, numerical expression, numerical expression may be hidden.

Before wondering how to solve algebraic fractions, first need to understand that such an expression is part of the whole.

As a rule, the whole is 1. The number in the denominator shows how many parts were divided by a unit. The numerator is necessary in order to find out how many elements are taken. The fractional feature corresponds to the division sign. It is allowed to record a fractional expression as a mathematical operation "Decision". In this case, the numerator is divisible, denominator - divider.

Major rule of ordinary fractions

When students take this topic at school, they are given examples to consolidate. To solve them correctly and find various ways from sophisticated situations, It is necessary to apply the basic property of fractions.

It sounds like this: if you multiply the numerator, and the denominator on the same number or expression (different from zero), then the value ordinary fraci Will not change. A special case of this rule is the separation of both parts of the expression on the same number or polynomial. Such transformations are called identical equalities.

Below will be considered how to solve the addition and subtraction of algebraic fractions, to produce multiplication, division and reduction of fractions.

Mathematical transactions with fractions

Consider how to solve, the main property of algebraic fraction, how to apply it in practice. If you need to multiply two fractions, fold them, divide one to another or deduct, you must always stick to the rules.

So, for the addition operation and subtraction, an additional factor should be found to bring expressions to the general denominator. If initially fractions are given with the same q expressions, then you need to lower this item. When common denominator Found how to solve algebraic fractions? You need to fold or subtract numerals. But! It must be remembered that if there is a sign "-" before the fraction, all signs in the numener are changing to the opposite. Sometimes you should not make any substitutions and mathematical operations. Enough to change the sign before the fraction.

Often used such a thing as reducing fractions. This means the following: if the numerator and the denominator are divided into an expression other than the unit (the same for both parts), then a new fraction is obtained. The divider and divider is less than the former, but due to the basic rules of fractions remain equal in the original example.

The purpose of this operation is to obtain a new non-interpretable expression. You can solve this task if you cut the numerator and the denominator to the largest general divisor. The operation algorithm consists of two points:

1. Finding a node for both parts of the fraction.
2. The division of the numerator and the denominator for the found expression and the receipt of an unstable fraction equal to the previous one.

Below is the table in which the formulas are painted. For convenience, it can be printed and carry with you in the notebook. However, in order to solve the control or exam in the future in the future, there was no difficulty solving how to solve algebraic fractions, these formulas need to be learned by heart.

Some examples with solutions

From the theoretical point of view, the question of how to solve algebraic fractions. Examples given in the article will help better learn the material.

1. Transform fractions and lead them to a common denominator.

2. Convert fractions and lead them to a common denominator.

After studying the theoretical part and the search for practical issues should not be more.