Examples Reduce algebraic fraction. Transformation of expressions

Before proceeding to the study algebraic fractions We recommend to remember how to work with ordinary fractions.

Any fraction in which there is an alphabetic factor is called an algebraic fraction.

Examples algebraic fractions.

As with the ordinary fraction, in the algebraic fraction there is a numerator (upstairs) and denominator (below).

Reducing algebraic fractions

Algebraic fraction can be reduced. With reduction, use the rules for reducing ordinary fractions.

We remind you that with a reduction in ordinary fraction we also divided the numerator, and the denominator for the same number.

Algebraic fraction is reduced in the same way, but only the numerator and the denominator are divided into the same polynomial.

Consider an example of a reduction in algebraic fraction.

We define the smallest degree in which the "A" stands stand. The smallest degree for single-wing "A" is in the denominator - this is the second degree.

We divide, and the numerator, and the denominator on "A 2". When dividing homorals, use the property of the degree of private.

We remind you that any letter or number in a zero degree is a unit.

No need to write in detail each time, to which the algebraic fraction was reduced. It is enough to keep in mind the degree to which we reduced and record only the result.

A summary of the reduction of algebraic fraction looks like this.

Only the same letter factors can be cut.

Cannot cut

Can be cut

Other examples of reduction of algebraic fractions.

How to cut the fraction with polynomials

Consider another example of algebraic fraction. It is required to reduce the algebraic fraction, which in the numerator is worth a polynomial.

Reduce the polynomial in brackets can be only with exactly the same polynomial in brackets!

In no case you can not cut part The polynomial inside the brackets!

Wrong

Determine where the polynomial ends is very simple. There can be only a sign of multiplication between polynomials. The entire polynomial is inside the brackets.

After we have identified polynomials of algebraic fractions, reduce the polynomial "(m - n)" in a numerator with a polynomial "(M - n)" in the denominator.

Examples of reduction of algebraic fractions with polynomials.

Reaching a common factor when cutting fractions

In order for in algebraic fractions, the same polynomials sometimes need to make a common factor for brackets.

In this form, it is impossible to reduce the algebraic fraction, since the polynomial
"(3f + k)" can only be reduced with a polynomial "(3f + k)".

Therefore, in order to obtain "(3f + k) in the numerator," I will summarize the "5" multiplier.

Reducing fractions using the formulas of abbreviated multiplication

In other examples, to reduce algebraic fractions required
applying formulas of abbreviated multiplication.

In the initial form, it is impossible to reduce the algebraic fraction, since there are no identical polynomials.

But if you apply the formula for the difference in the squares for the polynomial "(A 2 - B 2)", then the same polynomials will appear.

Other examples of the reduction of algebraic fractions using the formulas of abbreviated multiplication.

Reducing algebraic (rational) fractions is based on their main property: if the numerator and denominator are divided into the same nonzero polynomial, then the fraction equal to it.

Only multipliers can be cut!

Members of the polynomials cannot be cut!

To reduce the algebraic fraction, the polynomials standing in the numerator and the denominator must first decompose on multipliers.

Consider the examples of the reduction of fractions.

In the numerator and denominator, the frarators are classified. They represent composition (numbers, variables and their degrees), multipliers We can cut.

The numbers reduce their largest common divisor, that is, the largest number to which each of these numbers is divided. For 24 and 36, it is 12. After the reduction of 24 remains 2, from 36 - 3.

Degree reduce to the degree with the smallest indicator. Reduce the fraction means to divide the numerator and the denominator to the same divider, and when degree of degrees, we subtract the indicators.

a² and A⁷ reducing A². At the same time, a unit remains in a numerator from A² (1 write only in the case, when it is left, after the reduction of other factors, it remained. From 24 remained 2, therefore 1 remaining from A², do not write). From A⁷ after the reduction remains A⁵.

b and B reducing on B, the resulting units do not write.

c³º and Sling on S⁵. From C³º remains C² ⁵, from C⁵ - one (do not write it). In this way,

Numerator and denominator of this algebraic fraction - polynomials. Cut the members of the polynomials can not! (Cannot cut, for example, 8x² and 2x!). To reduce this fraction, it is necessary to decompose the polynomials on multipliers. The numerator has a total multiplier 4x. We carry it out for brackets:

Both in the numerator, and in the denominator there is the same multiplier (2x-3). Reduce the fraction in this multiplier. In the numerator received 4x, in the denominator - 1. According to 1 property of algebraic fractions, the fraction is 4x.

Only multipliers can be cut (it is impossible to reduce this fraction on 25x²!). Therefore, the polynomials standing in the numerator and the denomoter of the fraction should be decomposed on multipliers.

In the numerator - the full square of the amount, in the denominator - the difference of squares. After decomposition according to the formulas of abbreviated multiplication, we get:

We reduce the fraction on (5x + 1) (for this, in the numerator, you will cross the deuce in the indicator, from (5x + 1) ² will remain (5x + 1)):

In the numerator there is a general multiplier 2, I will bring it out of brackets. In the denominator - cube difference formula:

As a result of the decomposition in the numerator and the denominator, the same multiplier was obtained (9 + 3a + a²). Reduce the fraction on it:

The polynomial in the numerator consists of 4 terms. We group the first term with the second, the third - with the fourth and endure from the first brackets, the total multiplier X². The denominator is expanding according to the formula of the cubes:

In the numerator, we submit a general multiplier for brackets (X + 2):

We reduce the fraction on (x + 2):

Only multipliers can cut! To reduce this fraction, you need to decompose polynomials in the numerator and denominator. In the numerator, the total multiplier A³, in the denominator - A⁵. Let's take them for brackets:

Multipliers - degrees with the same base A³ and A⁵ - reducing on a³. From a³ remains 1, we do not write it, from A⁵ remains A². In the numerator, the expression in brackets can be decomposed as a difference of squares:

We reduce the fraction on the general divider (1 + a):

And how to cut the fraction of the species

in which the expression standing in the numerator and the denominator differ only on signs?

Examples of reduction of such fractions We will consider the next time.

2 comments

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Reducing algebraic frains Rule

Reducing algebraic fractions

A new concept in mathematics rarely arises from nothing "," on an empty place. " It appears when it feels an objective necessity. That is how negative numbers appeared in mathematics, so ordinary and decimal algebraic fraci.

Prerequisites for the introduction of a new concept "Algebraic fraction" we have. Let's lend to § 12. Discussing the division there is unoblared on one-time, we reviewed a number of examples. We highlight two of them.

1. To divide the one-wing 36a 3 B 5 per single-wing 4ab 2 (see Example 1B) from §12).
We solve it so. Instead of recording 36a 3 B 5: 4Ab 2 used fractions:

This allowed instead of the records 36: 4, and 3: a, b 5: b 2 also use the trait of the fraction, which made the solution of the example more visual:

2. To divide the single 4x 3 per single 2h (see example 1 d) from § 12). Acting on the same pattern, we got:

In § 12, we noted that 4x 3 was unintently. It was not possible to divide on one-time 2h so that it turned out monomial. But mathematical models Real situations may contain the operation of dividing any single-wing, not necessarily such that one is divided into another. Anticipating this, mathematics introduced a new concept - the concept of algebraic fraction. In particular, the algebraic fraction. Now let's go back to § 18. Discussing there the operation of division of the polynomial on the unrochene, we noted that it is not always done. So, in example 2 of § 18, it was about dividing twentiethly 6x 3 - 24x 2 on one-time 6x 2. This operation turned out to be performed and as a result we received twisted x - 4. So In other words, an algebraic expression managed to replace a simpler expression - a polynomial x - 4.

At the same time, in example 3 of § 18, the polynomial 8a 3 + ba 2b - b was divided into 2a 2, i.e. the expression could not be replaced by a simpler expression, it was necessary to leave it as an algebraic fraction.

As for the polynomial division operation polynomial, we actually did nothing about her. The only thing we can say now is: one polynomial can be divided into another if this other polynomial is one of the multipliers in the decomposition of the first polynomial to multipliers.

For example, x 3 - 1 \u003d (x - 1) (x 2 + x + 1). So x 3 - 1 can be divided by x 2 + x + 1, it turns out x - 1; x 3 - 1 can be divided by x - 1,

it turns out x 2 + x + 1.
polynomials P and Q. At the same time use recording
where p is a numerator, q - denominator of algebraic fraction.
Examples of algebraic fractions:

Sometimes an algebraic fraction can be replaced by a polynomial. For example, as we have already installed earlier,

(polynomial 6x 3 - 24x 2 managed to divide by 6x 2, while in particular it turns out x - 4); We also noted that

But it is relatively rare.

However, a similar situation has already met you - when studying ordinary fractions. For example, the fraction can be replaced with an integer 4, and the fraction is an integer 5. However, the fraction cannot be replaced with an integer, although this fraction can be reduced by separating the numerator and the denominator to the number 8 - the total multiplier of the numerator and the denominator:
In the same way, you can shorten the algebraic fractions, dividing the numerator and denominator of the fraction on their common mew. And for this you need to decompose and the numerator, and the denomoter of the factors. Here we will need all that we have discussed in this chapter so long.

Example. Reduce algebraic fraction:

Solution, a) We will find a general factor for homorals
12x 3 in 4 and 8x 2 in 5 as we did in § 20. We get 4x 2 in 4. Then 12x 3 y 4 \u003d 4x 2 y 4 sq; 8x 2 y 5 \u003d 4x 2 y 4 2y.
It means

Numerator I. denominator The given algebraic fraction has reduced the total multiplier of 4x 2 in 4.
The solution of this example can be recorded differently:

b) To shorten the fraction, spread its numerator and denominator for multipliers. We get:

(The fraction was reduced to the general factor a + b).

And now return to the remark 2 of § 1. See, we finally managed this promise there.
c) we have:

(reduced the fraction on the general factor of the numerator and the denominator, i.e. on x (x - y))

So, in order to reduce the algebraic to the fraction, it is necessary first of all to decompose its numerator and denominator. So your success in this new business (reduction of algebraic fractions) mainly depends on how you learned the material of previous paragraphs of this chapter.

A. V. Pogorelov, Geometry for 7-11 classes, Textbook for general education institutions

If you have corrections or suggestions for this lesson, write to us.

If you want to see other adjustments and wishes to the lessons, see here - Educational Forum.

Reducing algebraic fractions: rule, examples.

We continue to study the topic of the transformation of algebraic fractions. In this article we will focus in detail on reducing algebraic fractions. First, we will understand what they understand the term "reducing algebraic fraction", and find out whether the algebraic fraction is always reduced. Next, we give the rule to allow this conversion. Finally, we consider solutions of characteristic examples that will allow to understand all the subtleties of the process.

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What does it mean to reduce algebraic fraction?

Studying ordinary fractionsWe talked about their reduction. With a reduction in ordinary fraction, we called the division of its number and denominator to the general factory. For example, an ordinary fraction of 30/54 can be reduced by 6 (that is, divided into 6 its numerator and denominator), which will lead us to the fraction 5/9.

Under the reduction of algebraic fractions understand a similar effect. Reduce algebraic fraction - It means splitting its numerator and denominator to a general factor. But if the common factory of the numerator and denominator of the ordinary fraction can be only a number, then the general factor of the numerator and denominator of the algebraic fraction can be polynomial, in particular, single or number.

For example, algebraic fraction can be reduced by number 3 that will give a fraction . You can also reduce the variable x, which will lead to expression . The initial algebraic fraction can be reduced to single-wing 3 · x, as well as on any of the polynomials x + 2 · y, 3 · x + 6 · y, x 2 + 2 · x · y or 3 · x 2 + 6 · x · y.

The ultimate goal of the reduction of algebraic fraction consists in obtaining a fraction of a simpler view, at best, an unstable fraction.

Is any algebraic fraction to be reduced?

We know that ordinary fractions are divided into shortened and non-constructed fractions. Unstable fractions do not have different from the unit of common multipliers in a numerator and denominator, therefore, are not subject to reduction.

Algebraic fractions may also have common multipliers of the numerator and denominator, and may not have. If there are general factors, there is a reduction in algebraic fraction. If there are no general factors, then simplification of algebraic fraction is impossible through its reduction.

In general, according to the appearance of the algebraic fraction, it is rather difficult to determine whether it is possible to accumulate it. Undoubtedly, in some cases, general multipliers of the numerator and denominator are obvious. For example, it is clearly seen that the numerator and denominator of the algebraic fraction have a general multiplier 3. It is also easy to notic that the algebraic fraction can be reduced by x, on y or immediately to x · y. But much more often than the general factor of the numerator and the denominator algebraic fraction is not immediately visible, and more often - it is simply not. For example, the fraction can be reduced by X-1, but this common factor is clearly not present in the record. And algebraic fraction it is impossible to reduce, since its numerator and denominator do not have common multipliers.

In general, the question of the reduction of algebraic fraction is very difficult. And sometimes it is easier to solve the task, working with an algebraic fraction in its original form than to find out if this fraction can be pre-reduced. But still there are transformations, which in some cases allow with relatively minor efforts to find common multipliers of the numerator and the denominator, if any, or conclude the inconsistence of the initial algebraic fraction. This information will be disclosed in the next paragraph.

The rule of reduction of algebraic fractions

Information of previous paragraphs allows you to naturally perceive the following the rule of reduction of algebraic fractionswhich consists of two steps:

• first there are general multipliers of the numerator and denominator of the original fraction;
• if any, then there is a reduction in these multipliers.

These steps of the voiced rule need clarification.

The most convenient way of finding the general is to decompose the multi-polynomials in the numerator and denominator of the original algebraic fraction. At the same time, general multipliers of the numerator and the denominator become visible, or it becomes clear that there are no general factors.

If there are no general multipliers, we can conclude that the algebraic fraction is not constructed. If the general factors are found, then in the second step they are reduced. As a result, a new fraction of a simpler view is obtained.

The rule of reduction of algebraic fractions is based on the basic property of algebraic fraction, which is expressed by equality, where A, B and C are some polynomials, with B and C - nonzero. In the first step, the initial algebraic fraction is given to the form, from which the general multiplier C becomes visible, and in the second step, the reduction is performed - the transition to the fraction.

Go to solving examples using this rule. We will analyze all possible nuances that arise when decomposing the numerator and denominator of algebraic fractions on multipliers and subsequent reduction.

Characteristic examples

First you need to say about the reduction of algebraic fractions, the numerator and denominator of which are the same. Such fractions are identically equal to one throughout the EDD of the variables included in it, for example,
etc.

Now it will not hurt to remember how the reduction of ordinary fractions is carried out - after all, they are a special case of algebraic fractions. Natural numbers in a numerator and a denominator of ordinary fraci are colorful to simple multipliers, after which total multipliers are reduced (if available). For example, . The product of the same simple multipliers can be recorded in the form of degrees, and while reducing the property of degree in degrees with the same bases. In this case, the solution would look like this: Here we are a numerator and a denominator divided into a general multiplier 2 2 · 3. Or for greater clarity on the basis of the properties of multiplication and division, the solution is represented in the form.

In absolutely similar principles, the algebraic fractions are reduced, in the numerator and denominator of which are unknown with integer coefficients.

Reduce algebraic fraction .

It is possible to represent the numerator and denominator of the original algebraic fraction as a product of simple multipliers and variables, after which it is reduced:

But the solution is more rationally write in the form of expressions with degrees:

.

As for the reduction of algebraic fractions with fractional numerical coefficients in a numerator and denominator, it is possible to flow two: either separately perform the division of these fractional coefficients, or to pre-get rid of fractional coefficients, multiplying the numerator and denominator for some natural number. We talked about the last transformation in the article, bringing algebraic fractions to a new denominator, it can be carried out by virtue of the basic properties of algebraic fraction. We will deal with this on the example.

Perform a cutting of the fraction.

You can cut the fraction as follows: .

And it was possible to pre-get rid of fractional coefficients, multiplying the numerator and denominator to the smallest general multiple denominator of these coefficients, that is, on the NOC (5, 10) \u003d 10. In this case we have .

.

You can go to algebraic fractions general viewin which in the numerator and denominator can be both numbers and single, and polynomials.

With a reduction in such 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 general multiplier or make sure that it is not necessary for the numerator and denominator of algebraic fraction to decompose on multipliers.

Reduce the rational fraction .

To do this, we will decompose polynomials in a numerator and denominator. Let's start with the submission of the brackets :. Obviously, the expressions in brackets can be converted using the formulas of abbreviated multiplication: . Now it is clearly seen that it is possible to reduce the fraction on a common factor B 2 · (A + 7). Let's do it .

A brief solution without explanation is usually written in the form of a chain of equalities:

.

Sometimes general multipliers can be hidden by numeric coefficients. Therefore, with a reduction in rational fractions, numerical multipliers with senior degrees of the numerator and the denominator will be taken out for braces.

Reduce fraction , if possible.

At first glance, the numerator and denominator do not have a common factor. But still, let's try to perform some conversions. First, it is possible to make a multiplier X in a numerator: .

Now some similarity of expressions in brackets and expressions in the denominator due to x 2 · y are being blocked. I will bring numerical coefficients for the bracket with senior degrees of these polynomials:

After the transformation done, the general factory is visible, to which we carry out a reduction. Have

.

Completing the conversation about the reduction of rational fractions, we note that success depends on the ability to spread polynomials to multipliers.

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Reducing algebraic fractions

Relying on the above property, we can simplify algebraic fractions as well as it is done with arithmetic fractions, reducing them.

Reducing fractions is that the numerator and denominator of the fraction shall be divided into the same number.

If the algebraic fraction is unknown, the numerator and the denominator seems to be in the form of a product of several factors, and immediately can be seen which the same numbers can be divided into:

The same fraction we can write more details:. We see that you can consistently divide and the numerator and the denominator 4 times on A, i.e., in the end, divide each of them to a 4. Therefore ; Also, so on. So, if there are multipliers in the numerator and denominator, there are multiple degrees of the same letter, you can reduce this fraction to a smaller degree of this letter.

If the fraction is a polynomialist, then you have to first decompose these polynomials, if possible, for multipliers, and then the opportunity to see what the same multipliers can be divided into a numerator and denominator.

.... The numerator is easily folded on the factors "according to the formula" - it represents the square of the difference of two numbers, namely (x - 3) 2. The denominator is not suitable for formulas and will have to decompose it with a reception used for square three declared: raising 2 numbers, so that their sum is -1 and their product \u003d -6, - these numbers are -3 and + 2; Then x 2 - x - 6 \u003d x 2 - 3x + 2x - 6 \u003d x (x - 3) + 2 (x - 3) \u003d (x - 3) (x + 2).

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In this article we will analyze in detail how it is held reducing fractions. First, we will discuss what the fraction is called a reduction. After that, let's talk about bringing a reduced fraction to an incomprehensive form. Further, we will get a rule of reduction of fractions and, finally, consider examples of applying this rule.

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What does shorten the fraction mean?

We know that ordinary fractions are divided into reduced and non-constructed fractions. By names, it is possible to guess that the reduced fraction can be reduced, and non-conscript - it is impossible.

What does shorten the fraction mean? Reduce fraction - It means splitting its numerator and a denominator on their positive and different from one. It is clear that as a result of the reduction of the fraction, a new fraction with a smaller number and denominator is obtained, and, by virtue of the basic properties of the fraction, the resulting fraction is equal to the source.

For example, we will reduce the ordinary fraction 8/24, separating its numerator and denominator to 2. In other words, we will reduce the fraction 8/24 to 2. Since 8: 2 \u003d 4 and 24: 2 \u003d 12, as a result of such a reduction, it turns out a fraction 4/12, which is equal to the initial fraction 8/24 (see equal and unequal fractions). In the end we have.

Bringing ordinary fractions to nonstorm

Usually the ultimate goal of the reduction of the fraction is to obtain a non-interpretable fraction, which is equal to the initial reduced fraction. This goal can be achieved if it is reduced by the initial reduced fraction on its numerator and denominator. As a result of such a reduction, an unstable fraction is always obtained. Indeed, fraction is non-worn, because from it is known that and -. Here, let's say that the greatest common divisor of the numerator and denominator of the fraction is the greatest number that can be reduced by this fraction.

So, bringing ordinary fractions to an incomprehensive form It is to divide the numerator and denominator of the initial reduced fraction on their node.

We will analyze an example, for which we will return to the fraction 8/24 and reduce it to the largest common divisor of numbers 8 and 24, which is 8. Since 8: 8 \u003d 1 and 24: 8 \u003d 3, then we arrive at the non-interpretable fraction 1/3. So, .

Note that under the phrase "cut a fraction" often implies the leading of the initial fraction precisely to an incomprehensive form. In other words, the cutting of the fraction is very often called the division of the numerator and the denominator on their greatest common divisor (and not on any of their common divisor).

How to cut a fraction? Rule and fraction reduction examples

It remains only to disassemble the shortage of fractions, which explains how to reduce this fraction.

The reduction rule of fractions Consists of two steps:

• first, there is a node of the numerator and denominator of the fraction;
• secondly, the division of the numerator and the denominator of the fraction on their nodes is carried out, which gives an incomprehensive fraction equal to the initial one.

We will understand an example of a reduction of the fraci According to the voiced rule.

Example.

Reduce fraction 182/195.

Decision.

We carry out both steps prescribed by the rules of the cutting of the fraction.

First we find Nod (182, 195). It is most convenient to use the Euclide algorithm (see): 195 \u003d 182 · 1 + 13, 182 \u003d 13 · 14, that is, node (182, 195) \u003d 13.

Now we divide the numerator and denominator of the fraction 182/195 by 13, while we get an incompreheral fraction 14/15, which is equal to the initial fraction. On this cutting of the fraction is completed.

Briefly the solution can be written like this :.

Answer:

On this with a reduction of fractions, it is possible to finish. But for the completeness of the picture, consider two more ways to reduce fractions, which are usually applied in easy cases.

Sometimes the numerator and denominator of the cutting fraction is easy. Reduce the fraction in this case is very simple: you only need to remove all common multipliers from the numerator and denominator.

It is worth noting that this method directly follows from the rule of reduction of fractions, since the product of all common simple multipliers of the numerator and the denominator is equal to their greatest general divisor.

We will analyze the solution of the example.

Example.

Reduce fraction 360/2 940.

Decision.

Spread the nipple and denominator for simple multipliers: 360 \u003d 2 · 2 · 2 · 3 · 3 · 5 and 2 940 \u003d 2 · 2 · 3 · 5 · 7 · 7. In this way, .

Now we get rid of general multipliers in the numerator and denominator, for convenience, they simply cry out: .

Finally, I turn out the remaining multipliers:, and the reduction of the fraction is completed.

Here is a brief record of the decision: .

Answer:

Consider another way to reduce the fraction, which consists in a consistent reduction. Here at each step there is a reduction in the fraction on some common divisor of the numerator and denominator, which is either obvious or easily determined by

This article continues the topic of transformation of algebraic fractions: Consider such an action as a reduction in algebraic fractions. Let us give the definition of the term itself, we formulate the reduction rule and analyze practical examples.

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The meaning of the reduction of algebraic fraction

In the materials on ordinary fraction, we considered its reduction. We have determined the reduction of ordinary fraction as a division of its number and denominator for a common factor.

Reducing the algebraic fraction is a similar action.

Definition 1.

Reducing algebraic fractions - This is the division of its numerator and denominator for a general factor. At the same time, in contrast to the reduction of an ordinary fraction (the total denominator can only be a number), the total multiplier of the numerator and denominator of the algebraic fraction can serve as a polynomial, in particular, or a number.

For example, the algebraic fraction 3 · x 2 + 6 · x · y 6 · x 3 · y + 12 · x 2 · y 2 can be reduced by number 3, as a result, we obtain: x 2 + 2 · x · y 6 · x 3 · y + 12 · x 2 · y 2. We can cut the same fraction to the variable x, and it will give us the expression 3 · x + 6 · y 6 · x 2 · y + 12 · x · y 2. Also a given fraction can be reduced by one-sided 3 · X.or any of the polynomials X + 2 · Y, 3 · x + 6 · y, x 2 + 2 · x · y or 3 · x 2 + 6 · x · y.

The ultimate goal of the reduction of algebraic fraction is the fraction of a simpler point, at best, an unstable fraction.

Are all algebraic fractions subject to reduction?

Again, from materials on ordinary fractions, we know that there are cuts and non-interpretable fractions. Unstable is a fraction who do not have common multipliers of the numerator and denominator different from 1.

With algebraic fractions, everything is the same: they may have common multipliers of the numerator and denominator, may not have. The presence of general factors makes it possible to simplify the initial fraction by reducing. When there are no general multipliers, it is impossible to optimize the specified fraction of the reduction.

In general cases, according to the specified type, the fraction is quite difficult to understand whether it is subject to a reduction. Of course, in some cases, the presence of a common multiplier of the numerator and denominator is obvious. For example, in algebraic fractions 3 · x 2 3 · y, it is absolutely clear that the total factor is the number 3.

In the fraction - x · y 5 · x · y · z 3 We also immediately understand that it is possible to reduce it on x, or y, or on x · y. And yet, it is much more common examples of algebraic fractions, when the general multiplier of the numerator and the denominator is not so easy to see, and even more often - he is simply absent.

For example, the fraction of x 3 - 1 x 2 - 1 we can cut on x - 1, while the specified general multiplier in the record is missing. But the fraction x 3 - x 2 + x - 1 x 3 + x 2 + 4 · x + 4 is impossible to expose the reduction, since the numerator and the denominator do not have a common factor.

Thus, the question of finding out the reduction of algebraic fraction is not as simple, and it is often easier to work with the fraction of a given species than trying to figure out whether it is reduced. At the same time, there are such transformations that in particular cases allow you to determine the total multiplier of the numerator and the denominator or to conclude the fragility of the fraction. We will analyze in detail this question in the next paragraph of the article.

The rule of reduction of algebraic fractions

The rule of reduction of algebraic fractions consists of two consecutive actions:

• finding common multipliers of the numerator and denominator;
• if such, the implementation of the cutting effect of the fraction is directly.

The most convenient method of finding common denominators is the decomposition of polynomials existing in the numerator and denominator of a given algebraic fraction. This allows you to immediately see the presence or absence of general multipliers.

The effect of the reduction of algebraic fraction is based on the main property of an algebraic fraction expressed by the equality undefined, where a, b, C is some polynomials, and B and C - non-zero. The first step, the fraction is given to the form A · C B · C, in which we immediately notice the general factor c. The second step is to reduce, i.e. Transition to fraction of the form a b.

Characteristic examples

Despite some evidence, we clarify about a particular case when the numerator and denominator of algebraic fraction are equal. Similar fractions are identically equal to 1 throughout the odd variable of this fraction:

5 5 \u003d 1; - 2 3 - 2 3 \u003d 1; x x \u003d 1; - 3, 2 · x 3 - 3, 2 · x 3 \u003d 1; 1 2 · x - x 2 · y 1 2 · x - x 2 · y;

Since ordinary fractions are a special case of algebraic fractions, we will remind you how to reduce them. Natural numbers recorded in a numerator and denominator are laid out to simple multipliers, then general factors are reduced (if any).

For example, 24 1260 \u003d 2 · 2 · 2 · 3 2 · 2 · 3 · 3 · 5 · 7 \u003d 2 3 · 5 · 7 \u003d 2 105

The work of simple identical factors can be written as degrees, and in the process of reducing the fraction to use the property of degree in degrees with the same bases. Then the above decision would be:

24 1260 \u003d 2 3 · 3 2 2 · 3 2 · 5 · 7 \u003d 2 3 - 2 3 2 - 1 · 5 · 7 \u003d 2 105

(Numerator and denominator are divided into a common factor 2 2 · 3). Or for clarity, relying on the properties of multiplication and division, we will give this type of decision:

24 1260 \u003d 2 3 · 3 2 2 · 3 2 · 5 · 7 \u003d 2 3 2 2 · 3 3 2 · 1 5 · 7 \u003d 2 1 · 1 3 · 1 35 \u003d 2 105

By analogy, the algebraic fractions are reduced, in which the numeric and the denominator have universal with integer coefficients.

Example 1.

The algebraic fraction is given - 27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · C 7 · Z. It is necessary to make it reduced.

Decision

It is possible to write a numerator and denominator of a given fraction as a product of simple multipliers and variables, after which the reduction:

27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · C 7 · z \u003d - 3 · 3 · 3 · a · a · a · a · a · b · b · c · z 2 · 3 · A · A · b · b · C · C · C · C · C · C · C · Z \u003d \u003d - 3 · 3 · A · A · A 2 · C · C · C · C · C · C \u003d - 9 · a 3 2 · C 6

However, a more rational way will record a solution in the form of expressions with degrees:

27 · a 5 · b 2 · C · Z 6 · A 2 · B 2 · C 7 · Z \u003d - 3 3 · A 5 · B 2 · C · Z 2 · 3 · A 2 · B 2 · C 7 · z \u003d - 3 3 2 · 3 · a 5 a 2 · b 2 B 2 · Cc 7 · zz \u003d \u003d - 3 3 - 1 2 · a 5 - 2 1 · 1 · 1 C 7 - 1 · 1 \u003d · - 3 2 · a 3 2 · C 6 \u003d · - 9 · a 3 2 · C 6.

Answer: - 27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · C 7 · z \u003d - 9 · a 3 2 · C 6

When algebraic fraction in the numerator and denominator, there are fractional numerical coefficients, there are two ways of further actions: or separately divide these fractional coefficients, or to pre-get rid of fractional coefficients, multiplying the numerator and denominator for a kind of natural number. The last transformation is carried out due to the basic properties of the algebraic fraction (it is possible to read about it in the article "Running an algebraic fraction for a new denominator").

Example 2.

The fraction 2 5 · x 0, 3 · x 3 is given. It is necessary to reduce it.

Decision

It is possible to reduce the fraction in this way:

2 5 · x 0, 3 · x 3 \u003d 2 5 3 10 · x x 3 \u003d 4 3 · 1 x 2 \u003d 4 3 · x 2

Let us try to solve the problem otherwise, pre-getting rid of fractional coefficients - multiply the numerator and denominator to the smallest general multiple denominators of these coefficients, i.e. on NOC (5, 10) \u003d 10. Then we get:

2 5 · x 0, 3 · x 3 \u003d 10 · 2 5 · x 10 · 0, 3 · x 3 \u003d 4 · x 3 · x 3 \u003d 4 3 · x 2.

Answer: 2 5 · x 0, 3 · x 3 \u003d 4 3 · x 2

When we reduce the algebraic fraction of a shared form, in which the numerals and denominators can be both universal and polynomials, a problem is possible when the general factor is not always visible immediately. Or moreover, he simply does not exist. Then, to determine the general factor or fixing the fact about its absence, the numerator and the denominator of the algebraic fraction lay out on multipliers.

Example 3.

The rational fraction 2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 is given. It is necessary to cut it.

Decision

We will decompose polynomials in a numerator and denominator. Implement for braces:

2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 \u003d 2 · b 2 · (A 2 + 14 · A + 49) B 3 · (A 2 - 49)

We see that the expression in brackets can be converted using the formulas of abbreviated multiplication:

2 · b 2 · (A 2 + 14 · A + 49) B 3 · (A 2 - 49) \u003d 2 · B 2 · (A + 7) 2 B 3 · (A - 7) · (A + 7)

It is clearly noticeable that it is possible to reduce the fraction on the general factory B 2 · (A + 7). We will reduce:

2 · b 2 · (A + 7) 2 B 3 · (A - 7) · (A + 7) \u003d 2 · (A + 7) B · (A - 7) \u003d 2 · A + 14 A · B - 7 · B.

A brief decision without explanation we write as a chain of equalities:

2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 \u003d 2 · b 2 · (A 2 + 14 A + 49) B 3 · (A 2 - 49) \u003d \u003d 2 · b 2 · (A + 7) 2 B 3 · (A - 7) · (A + 7) \u003d 2 · (A + 7) B · (A - 7) \u003d 2 · A + 14 A · b - 7 · b

Answer: 2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 \u003d 2 · A + 14 A · B - 7 · b.

It happens that common factors are hidden by numeric coefficients. Then, when cutting fractions, the optimal numerical factors with the senior degrees of the numerator and the denominator to take place behind the brackets.

Example 4.

Dana algebraic fraction 1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2. It is necessary to carry out its reduction, if possible.

Decision

At first glance, the numerator and denominator does not exist general denominator. However, let's try to convert a given fraction. I will bring a multiplier x in a numerator:

1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2 \u003d x · 1 5 - 2 7 · x 2 · y 5 · x 2 · y - 3 1 2

Now a certain similarity of expressions in brackets and expressions in the denominator due to x 2 · y . I will bring numerical coefficients for the bracket with senior degrees of these polynomials:

x · 1 5 - 2 7 · x 2 · y 5 · x 2 · y - 3 1 2 \u003d x · - 2 7 · - 7 2 · 1 5 + x 2 · y 5 · x 2 · y - 1 5 · 3 1 2 \u003d - - 2 7 · x · - 7 10 + x 2 · y 5 · x 2 · y - 7 10

Now the general multiplier becomes visible, we carry out a reduction:

2 7 · x · - 7 10 + x 2 · y 5 · x 2 · y - 7 10 \u003d - 2 7 · x 5 \u003d - 2 35 · x

Answer: 1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2 \u003d - 2 35 · x.

Let the emphasis on the fact that the skill of the reduction of rational fractions depends on the ability to spread polynomials to multipliers.

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At first glance, algebraic fractions seem very difficult, and an unprepared student may think that it is impossible to do anything with them. The journey of variables, numbers and even degrees imposes fear. Nevertheless, to reduce the usual (for example, 15/25) and algebraic fractions, the same rules are used.

Steps

Reducing fractions

Check out the actions with simple fractions. Operations with conventional and algebraic fractions are similar. For example, we take a shot 15/35. To simplify this fraction, follows find a common divider. Both numbers are divided by five, so we can highlight 5 in the numerator and denominator:

15 → 5 * 3 35 → 5 * 7

Now you can reduce general multipliers, that is, delete 5 in the numerator and denominator. As a result, we get a simplified fraction 3/7 . IN algebraic expressions Common multipliers stand out in the same way as in ordinary. In the previous example, we were able to easily distinguish 5 out of 15 - the same principle is applicable to more complex expressions, such as 15x - 5. We will find a general factor. In this case, it will be 5, since both members (15x and -5) are divided by 5. As before, we will highlight a common multiplier and transfer it left.

15x - 5 \u003d 5 * (3x - 1)

To check whether everything is correct enough to multiply 5 standing in brackets in brackets - as a result, the same numbers will be at first. Complex members can be allocated in the same way as simple. For algebraic fractions, the same principles apply as for ordinary ones. This is the easiest way to reduce the fraction. Consider the following fraction:

(x + 2) (X-3)(x + 2) (x + 10)

Note that in the numerator (from above), and in the denominator (bottom) there is a member (X + 2), so it can be reduced in the same way as the total multiplier 5 in the fraction 15/35:

(x + 2) (X-3) → (X-3) (x + 2) (x + 10) → (x + 10)

As a result, we obtain a simplified expression: (x-3) / (x + 10)

Reducing algebraic fractions

Find a common multiplier in a numerator, that is, in the upper part of the fraction. With the reduction of algebraic fractions, the first thing should simplify both parts of it. Start from the numerator and try to decompose it on as many factors as possible. Consider in this section the following fraction:

9x-3.15x + 6.

Let's start with the numerator: 9x - 3. For 9x and -3, the total factor is the number 3. I will summarize 3 per brackets, as is done with conventional numbers: 3 * (3x-1). As a result of this transformation, the next fraction will turn out:

3 (3x-1)15x + 6.

Find a common multiplier in the numerator. We will continue to execute the above example and drank the denominator: 15x + 6. As before, we will find what number both parts are divided. And in this case, the general factor is 3, so you can write: 3 * (5x +2). Let's rewrite the fraction in the following form:

3 (3x-1)3 (5x + 2)

Reduce the same members. At this step, you can simplify the fraction. Reduce the same members in a numerator and denominator. In our example, this is the number 3.

3 (3x-1) → (3x-1) 3 (5x + 2) → (5x + 2)

Determine that the fraction has the simplest view. The fraction is completely simplified in the case when there are no general multipliers in the numerator and denominator. Note that it is impossible to reduce those members that stand inside the brackets - in the example above, it is not possible to allocate x from 3x and 5x, since complete members are (3x -1) and (5x + 2). Thus, the fraction does not give in to further simplification, and the final answer is as follows:

(3x-1)(5x + 2)

Repeat cut fractions yourself. The best way to assimilate the method is independent decision Tasks. Under examples are given correct answers.

4 (x + 2) (X-13)(4x + 8)

Answer: (x \u003d 13)

2x 2 -X.5x

Answer:(2x-1) / 5

Special techniques

Take a negative sign beyond the fraction. Suppose, the next fraction is given:

3 (X-4)5 (4-x)

Note that (X-4) and (4-x) "almost" identical, but they cannot be reduced immediately, as they are "turned over". However, (x - 4) can be written as -1 * (4 - x), just as (4 + 2x) can be rewritten as 2 * (2 + x). This is called "Sign Change".

-1 * 3 (4-x)5 (4-x)

Now you can reduce the same members (4-x):

-1 * 3 (4-x)5 (4-x)

So, we get the final answer: -3/5 . Learn to recognize the difference in squares. The difference in squares is when the square of one number is subtracted from the square of another number, as in the expression (A 2 - B 2). The difference in full squares can always be decomposed into two parts - the amount and difference of the corresponding square roots. Then the expression will take the following form:

A 2 - B 2 \u003d (A + B) (A-B)

This technique is very useful when searching for general members in algebraic fractions.

• Check if you laid the correct expression on multipliers. To do this, multiply multipliers - as a result, the same expression should be obtained.
• To fully simplify the fraction, always allocate the greatest multipliers.

Subject:Depaction of polynomials for multipliers

Lesson:Algebraic fractions. Reducing algebraic fractions in more complex cases

Recall that algebraic is the attitude of polynomials:

In the previous lesson, we conducted an analogy between the algebraic fraction and arithmetic fraction. Recall:

The result of decomposition on the multipliers of the numerator and the denominator is some fraction;

Specifically it was a fraction

Sperate the specified expression:

We replace the number of changes in X, Y, Z, we get:

Recall that the main task when working with algebraic fractions is to decompose the numerator and denominator for multipliers and if such an opportunity to reduce general multipliers.

Consider examples:

We convert the numerator using the square difference formula:

Sperate the emerging general multiplier:

As a result of the division of bouquets, two-headeds were obtained, which we painted the formula of the Cubes difference and received its discontinuity on multipliers;

Spread the numerator and denominator on multipliers. The denominator is explicitly the formula of the square of the sum, and in the numerator under the square there is a difference in squares:

We will reveal the square in the numerator, for this, each multiplier is erected into the square:

Sperate the general factory:

Example 3 - Simplify the fraction and calculate its value when:

Spread the numerator and denominator on multipliers:

Sperate the general factory:

We will substitute the value and calculate the value of the fraci:

Example 4 - Simplify the fraction and calculate its value when:

Apply to the numerator the formula of the difference of squares, and to the denominator the sum of the sum of the square:

We will substitute the value and calculate:

Example 5 - decompose on multipliers:

Apply the method of grouping to decompose the number and denominator:

Sperate the general factory:

Output: In this lesson, we remembered what is an algebraic fraction and what the basics of working with it. We learned how to solve complex examples and secured the skills for solving tasks with algebraic fractions.

1. Dorofeyev G.V., Suvorova S.B., Baynovich E.A. And others. Algebra 7. 6 Edition. M.: Enlightenment. 2010

2. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebra 7. M.: Ventana Graph

3. Kolyagin Yu.M., Tkachev M.V., Fedorova N.E. and others. Algebra 7 .m .: Enlightenment. 2006

1. All elementary mathematics ().

Task 1: Kolyagin Yu.M., Tkachev M.V., Fedorova N.E. and others. Algebra 7, No. 446, Art.152;

Task 2: Kolyagin Yu.M., Tkachev M.V., Fedorova N.E. and others. Algebra 7, No. 447, Art.152;

Task 3: Kolyagin Yu.M., Tkachev M.V., Fedorova N.E. et al. Algebra 7, No. 448, Art.152;