In this article, we will focus on reduction of algebraic fractions. First, let's figure out what is meant by the term "reduction of an algebraic fraction", and find out whether an algebraic fraction is always reducible. Next, we give a rule that allows us to carry out this transformation. Finally, consider the solutions of typical examples that will make it possible to understand all the subtleties of the process.
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What does it mean to reduce an algebraic fraction?
Studying, we talked about their reduction. we called the division of its numerator and denominator by the common factor. For example, the common fraction 30/54 can be reduced by 6 (that is, divided by 6 its numerator and denominator), which will lead us to the fraction 5/9.
The reduction of an algebraic fraction is understood as a similar action. Reduce algebraic fraction is to divide its numerator and denominator by a common factor. But if the common factor of the numerator and denominator of an ordinary fraction can only be a number, then the common factor of the numerator and denominator of an algebraic fraction can be a polynomial, in particular, a monomial or a number.
For example, an algebraic fraction can be reduced by the number 3, which gives the fraction 
. It is also possible to reduce on the variable x , which will result in the expression 
. The original algebraic fraction can be reduced by the monomial 3 x, as well as by 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 reducing an algebraic fraction is to obtain a fraction of a simpler form, at best, an irreducible fraction.
Is any algebraic fraction subject to reduction?
We know that ordinary fractions are subdivided into . Irreducible fractions do not have common factors other than unity in the numerator and denominator, therefore, they cannot be reduced.
Algebraic fractions may or may not have common numerator and denominator factors. In the presence of common factors, it is possible to reduce the algebraic fraction. If there are no common factors, then the simplification of the algebraic fraction by means of its reduction is impossible.
In the general case, by the appearance of an algebraic fraction, it is quite difficult to determine whether it is possible to perform its reduction. Undoubtedly, in some cases the common factors of the numerator and denominator are obvious. For example, it is clearly seen that the numerator and denominator of an algebraic fraction have a common factor of 3. It is also easy to see that an algebraic fraction can be reduced by x, by y, or immediately by x·y. But much more often, the common factor of the numerator and denominator of an algebraic fraction is not immediately visible, and even more often, it simply does not exist. For example, a fraction can be reduced by x−1 , but this common factor is clearly not present in the notation. And an algebraic fraction 
cannot be reduced because its numerator and denominator do not have common factors.
In general, the question of the contractibility of an algebraic fraction is very difficult. And sometimes it’s easier to solve a problem by working with an algebraic fraction in its original form than to find out if this fraction can be preliminarily reduced. But still, there are transformations that in some cases allow, with relatively little effort, to find the common factors of the numerator and denominator, if any, or to conclude that the original algebraic fraction is irreducible. This information will be disclosed in the next paragraph.
Algebraic fraction reduction rule
The information of the previous paragraphs allows you to naturally perceive the following algebraic fraction reduction rule, which consists of two steps:
- first, the common factors of the numerator and denominator of the original fraction are found;
- if any, then reduction by these factors is carried out.
These steps of the announced rule need clarification.
The most convenient way to find common ones is to factorize the polynomials that are in the numerator and denominator of the original algebraic fraction. In this case, the common factors of the numerator and denominator immediately become visible, or it becomes clear that there are no common factors.
If there are no common factors, then we can conclude that the algebraic fraction is irreducible. If the common factors are found, then at the second step they are reduced. The result is a new fraction of a simpler form.
The rule of reduction of algebraic fractions is based on the main property of an algebraic fraction, which is expressed by the equality , where a , b and c are some polynomials, and b and c are non-zero. At the first step, the original algebraic fraction is reduced to the form , from which the common factor c becomes visible, and at the second step, reduction is performed - the transition to the fraction .
Let's move on to solving examples using this rule. On them, we will analyze all the possible nuances that arise when decomposing the numerator and denominator of an algebraic fraction into factors and subsequent reduction.
Typical 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 on the entire ODZ of the variables included in it, for example, 
and so on.
Now it does not hurt to remember how the reduction of ordinary fractions is performed - after all, they are a special case of algebraic fractions. Natural numbers in the numerator and denominator of an ordinary fraction, after which the common factors are reduced (if any). For example, 
. The product of identical prime factors can be written in the form of degrees, and when reduced, use. In this case, the solution would look like this: 
, here we divided the numerator and denominator by a common factor 2 2 3 . Or, for greater clarity, based on the properties of multiplication and division, the solution is presented in the form.
According to absolutely similar principles, the reduction of algebraic fractions is carried out, in the numerator and denominator of which there are monomials with integer coefficients.
Example.
Reduce algebraic fraction 
.
Solution.
You can represent the numerator and denominator of the original algebraic fraction as a product of simple factors and variables, and then carry out the reduction: 
But it is more rational to write the solution as an expression with powers: 
Answer:
.
As for the reduction of algebraic fractions that have fractional numerical coefficients in the numerator and denominator, you can do two things: either separately divide these fractional coefficients, or first get rid of the fractional coefficients by multiplying the numerator and denominator by some natural number. We talked about the last transformation in the article bringing an algebraic fraction to a new denominator, it can be carried out due to the main property of an algebraic fraction. Let's deal with this with an example.
Example.
Perform fraction reduction.
Solution.
You can reduce the fraction like this: 
.
And it was possible to get rid of fractional coefficients first by multiplying the numerator and denominator by the denominators of these coefficients, that is, by LCM(5, 10)=10 . In this case we have 
.
Answer:
.
You can move on to algebraic fractions of a general form, in which the numerator and denominator can contain both numbers and monomials, as well as polynomials.
When reducing such 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 make sure that it does not exist, you need to factorize the numerator and denominator of an algebraic fraction.
Example.
Reduce rational fraction 
.
Solution.
To do this, we factorize the polynomials in the numerator and denominator. Let's start with parentheses: . Obviously, parenthesized expressions can be converted using
We will understand what fraction reduction is, why and how to reduce fractions, we will give the rule for reducing fractions and examples of its use.
Yandex.RTB R-A-339285-1
What is "fraction reduction"
Reduce fractionTo reduce a fraction means to divide its numerator and denominator by a common divisor, positive and different from one.
As a result of such an action, a fraction with a new numerator and denominator will be obtained, equal to the original fraction.
For example, let's take the common fraction 6 24 and reduce it. Divide the numerator and denominator by 2, resulting in 6 24 = 6 ÷ 2 24 ÷ 2 = 3 12 . In this example, we have reduced the original fraction by 2 .
Reduction of fractions to irreducible form
In the previous example, we reduced the fraction 6 24 by 2 , resulting in the fraction 3 12 . It is easy to see that this fraction can be further reduced. Generally, the goal of reducing fractions is to end up with an irreducible fraction. How to convert a fraction to an irreducible form?
This can be done by reducing the numerator and denominator by their greatest common divisor (GCD). Then, by the property of the greatest common divisor, the numerator and denominator will be coprime numbers, and the fraction will be irreducible.
a b = a ÷ N O D (a , b) b ÷ N O D (a , b)
Reduction of a fraction to an irreducible form
To reduce a fraction to an irreducible form, you need to divide its numerator and denominator by their gcd.
Let's return to the fraction 6 24 from the first example and reduce it to an irreducible form. The greatest common divisor of 6 and 24 is 6 . Let's reduce the fraction:
6 24 = 6 ÷ 6 24 ÷ 6 = 1 4
Reducing fractions is convenient to use so as not to work with large numbers. In general, there is an unspoken rule in mathematics: if you can simplify any expression, then you need to do it. By reducing a fraction, most often they mean its reduction to an irreducible form, and not just reduction by a common divisor of the numerator and denominator.
Fraction reduction rule
To reduce fractions, it is enough to remember the rule, which consists of two steps.
Fraction reduction rule
To reduce a fraction:
- Find the gcd of the numerator and denominator.
- Divide the numerator and denominator by their gcd.
Consider practical examples.
Example 1. Let's reduce the fraction.
Given a fraction 182 195 . Let's shorten it.
Find the GCD of the numerator and denominator. For this, in this case, it is most convenient to use the Euclid algorithm.
195 = 182 1 + 13 182 = 13 14 N O D (182, 195) = 13
Divide the numerator and denominator by 13. We get:
182 195 = 182 ÷ 13 195 ÷ 13 = 14 15
Ready. We got an irreducible fraction, which is equal to the original fraction.
How else can you reduce fractions? In some cases, it is convenient to decompose the numerator and denominator into simple factors, and then remove all common factors from the upper and lower parts of the fraction.
Example 2. Reduce the fraction
Given a fraction 360 2940 . Let's shorten it.
To do this, we represent the original fraction in the form:
360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7
Let's get rid of the common factors in the numerator and denominator, as a result of which we get:
360 2940 = 2 2 2 3 3 5 2 2 3 5 7 7 = 2 3 7 7 = 6 49
Finally, consider another way to reduce fractions. This is the so-called sequential reduction. Using this method, the reduction is carried out in several stages, at each of which the fraction is reduced by some obvious common divisor.
Example 3. Reduce the fraction
Let's reduce the fraction 2000 4400 .
It is immediately clear that the numerator and denominator have a common factor of 100. We reduce the fraction by 100 and get:
2000 4400 = 2000 ÷ 100 4400 ÷ 100 = 20 44
20 44 = 20 ÷ 2 44 ÷ 2 = 10 22
The resulting result is again reduced by 2 and we get an irreducible fraction:
10 22 = 10 ÷ 2 22 ÷ 2 = 5 11
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Based on their main property: if the numerator and denominator of a fraction are divided by the same non-zero polynomial, then a fraction equal to it will be obtained.
You can only reduce multipliers!
Members of polynomials cannot be reduced!
To reduce an algebraic fraction, the polynomials in the numerator and denominator must first be factored.
Consider examples of fraction reduction.
The numerator and denominator of a fraction are monomials. They represent work(numbers, variables and their degrees), multipliers we can reduce.
We reduce the numbers by their greatest common divisor, that is, by the largest number by which each of the given numbers is divisible. For 24 and 36, this is 12. After the reduction from 24, 2 remains, from 36 - 3.
We reduce the degrees by the degree with the smallest indicator. To reduce a fraction means to divide the numerator and denominator by the same divisor, and subtract the exponents.
a² and a⁷ are reduced by a². At the same time, one remains in the numerator from a² (we write 1 only if, after reduction, there are no other factors left. From 24, 2 remains, so we do not write the 1 remaining from a²). From a⁷ after reduction remains a⁵.
b and b are abbreviated by b, the resulting units are not written.
c³º and c⁵ are reduced by c⁵. From c³º, c²⁵ remains, from c⁵ - unit (we do not write it). Thus,
The numerator and denominator of this algebraic fraction are polynomials. It is impossible to reduce the terms of polynomials! (cannot be reduced, for example, 8x² and 2x!). To reduce this fraction, it is necessary. The numerator has a common factor of 4x. Let's take it out of brackets:
Both the numerator and the denominator have the same factor (2x-3). We reduce the fraction by this factor. We got 4x in the numerator, 1 in the denominator. According to 1 property of algebraic fractions, the fraction is 4x.
You can only reduce factors (you cannot reduce a given fraction by 25x²!). Therefore, the polynomials in the numerator and denominator of a fraction must be factored.
The numerator is the full square of the sum, and the denominator is the difference of the squares. After expansion by the formulas of abbreviated multiplication, we get:
We reduce the fraction by (5x + 1) (to do this, cross out the two in the numerator as an exponent, from (5x + 1) ² this will leave (5x + 1)):
The numerator has a common factor of 2, let's take it out of brackets. In the denominator - the formula for the difference of cubes:
As a result of expansion in the numerator and denominator, we got the same factor (9 + 3a + a²). We reduce the fraction on it:
The polynomial in the numerator consists of 4 terms. the first term with the second, the third with the fourth, and we take out the common factor x² from the first brackets. We decompose the denominator according to the formula for the sum of cubes:
In the numerator, we take out the common factor (x + 2) out of brackets:
We reduce the fraction by (x + 2):
Children at school learn the rules for reducing fractions in 6th grade. In this article, we will first tell you what this action means, then we will explain how to translate a reducible fraction into an irreducible one. The next item will be the rules for reducing fractions, and then we will gradually get to the examples.
What does "reduce a fraction" mean?
So, we all know that ordinary fractions are divided into two groups: reducible and irreducible. Already by the names it can be understood that those that are contractible are reduced, and those that are irreducible are not reduced.
- To reduce a fraction is to divide its denominator and numerator by their (other than one) positive divisor. The result, of course, is a new fraction with a smaller denominator and numerator. The resulting fraction will be equal to the original fraction.
It is worth noting that in math books with the task "reduce the fraction", this means that you need to bring the original fraction to this irreducible form. In simple terms, dividing the denominator and numerator by their greatest common divisor is a reduction.
How to reduce a fraction. Rules for reducing fractions (Grade 6)
So there are only two rules here.
- The first rule for reducing fractions is to first find the greatest common divisor of the denominator and numerator of your fraction.
- Second rule: Divide the denominator and numerator by the greatest common divisor to end up with an irreducible fraction.
How to reduce an improper fraction?
The rules for reducing fractions are identical to the rules for reducing improper fractions.
In order to reduce an improper fraction, first you will need to paint the denominator and numerator into simple factors, and only then reduce the common factors.
Reduction of mixed fractions
The rules for reducing fractions also apply to the reduction of mixed fractions. There is only a small difference: we can not touch the whole part, but reduce the fractional or mixed fraction into an improper one, then reduce it and again convert it into a proper fraction.
There are two ways to reduce mixed fractions.
First: to paint the fractional part into prime factors and then do not touch the integer part.
The second way: first translate into an improper fraction, paint on the usual factors, then reduce the fraction. Convert the received improper fraction to the proper one.
Examples can be seen in the photo above.
We really hope that we could help you and your children. After all, in the classroom they are very often inattentive, so you have to work harder at home on your own.
Fractions
Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")
Fractions in high school are not very annoying. For the time being. Until you come across exponents with rational exponents and logarithms. And there…. You press, you press the calculator, and it shows all the full scoreboard of some numbers. You have to think with your head, like in the third grade.
Let's deal with fractions, finally! Well, how much can you get confused in them!? Moreover, it is all simple and logical. So, what are fractions?
Types of fractions. Transformations.
Fractions are of three types.
1. Common fractions , For example:
Sometimes, instead of a horizontal line, they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens ...), tell yourself the phrase with the expression: " Zzzzz remember! Zzzzz denominator - out zzzz u!" Look, everything will be remembered.)
A dash, which is horizontal, which is oblique, means division top number (numerator) to bottom number (denominator). And that's it! Instead of a dash, it is quite possible to put a division sign - two dots.
When the division is possible entirely, it must be done. So, instead of the fraction "32/8" it is much more pleasant to write the number "4". Those. 32 is simply divided by 8.
32/8 = 32: 8 = 4
I'm not talking about the fraction "4/1". Which is also just "4". And if it doesn’t divide completely, we leave it as a fraction. Sometimes you have to do the reverse. Make a fraction from a whole number. But more on that later.
2. Decimals , For example:
It is in this form that it will be necessary to write down the answers to tasks "B".
3. mixed numbers , For example:
Mixed numbers are practically not used in high school. In order to work with them, they must be converted to ordinary fractions. But you definitely need to know how to do it! And then such a number will come across in the puzzle and hang ... From scratch. But we remember this procedure! A little lower.
Most versatile common fractions. Let's start with them. By the way, if there are all sorts of logarithms, sines and other letters in the fraction, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!
Basic property of a fraction.
So let's go! First of all, I will surprise you. The whole variety of fraction transformations is provided by a single property! That's what it's called basic property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction will not change. Those:
It is clear that you can write further, until you are blue in the face. Do not let sines and logarithms confuse you, we will deal with them further. The main thing to understand is that all these various expressions are the same fraction . 2/3.
And we need it, all these transformations? And how! Now you will see for yourself. First, let's use the basic property of a fraction for fraction abbreviations. It would seem that the thing is elementary. We divide the numerator and denominator by the same number and that's it! It's impossible to go wrong! But... man is a creative being. You can make mistakes everywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.
How to reduce fractions correctly and quickly without doing unnecessary work can be found in special Section 555.
A normal student does not bother dividing the numerator and denominator by the same number (or expression)! He just crosses out everything the same from above and below! This is where a typical mistake lurks, a blunder, if you like.
For example, you need to simplify the expression:
There is nothing to think about, we cross out the letter "a" from above and the deuce from below! We get:
Everything is correct. But really you shared the whole numerator and the whole denominator "a". If you are used to just cross out, then, in a hurry, you can cross out the "a" in the expression
and get again
Which would be categorically wrong. Because here the whole numerator on "a" already not shared! This fraction cannot be reduced. By the way, such an abbreviation is, um ... a serious challenge to the teacher. This is not forgiven! Remember? When reducing, it is necessary to divide the whole numerator and the whole denominator!
Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. And how to work with her now? Without a calculator? Multiply, say, add, square!? And if you are not too lazy, but carefully reduce by five, and even by five, and even ... while it is being reduced, in short. We get 3/8! Much nicer, right?
The basic property of a fraction allows you to convert ordinary fractions to decimals and vice versa without calculator! This is important for the exam, right?
How to convert fractions from one form to another.
It's easy with decimals. As it is heard, so it is written! Let's say 0.25. It's zero point, twenty-five hundredths. So we write: 25/100. We reduce (divide the numerator and denominator by 25), we get the usual fraction: 1/4. All. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.
What if integers are non-zero? It's OK. Write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three whole, seventeen hundredths. We write 317 in the numerator, and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all of the above, a useful conclusion: any decimal fraction can be converted to a common fraction .
But the reverse conversion, ordinary to decimal, some cannot do without a calculator. And it is necessary! How will you write down the answer on the exam!? We carefully read and master this process.
What is a decimal fraction? She has in the denominator Always is worth 10 or 100 or 1000 or 10000 and so on. If your usual fraction has such a denominator, there is no problem. For example, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. And if in the answer to the task of section "B" it turned out 1/2? What will we write in response? Decimals are required...
We remember basic property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. For anyone, by the way! Except zero, of course. Let's use this feature to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But, then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 \u003d 1x5 / 2x5 \u003d 5/10 \u003d 0.5. That's all.
However, all sorts of denominators come across. For example, the fraction 3/16 will fall. Try it, figure out what to multiply 16 by to get 100, or 1000... Doesn't work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide in a corner, on a piece of paper, as they taught in elementary grades. We get 0.1875.
And there are some very bad denominators. For example, the fraction 1/3 cannot be turned into a good decimal. Both on a calculator and on a piece of paper, we get 0.3333333 ... This means that 1/3 into an exact decimal fraction does not translate. Just like 1/7, 5/6 and so on. Many of them are untranslatable. Hence another useful conclusion. Not every common fraction converts to a decimal. !
By the way, this is useful information for self-examination. In section "B" in response, you need to write down a decimal fraction. And you got, for example, 4/3. This fraction is not converted to decimal. This means that somewhere along the way you made a mistake! Come back, check the solution.
So, with ordinary and decimal fractions sorted out. It remains to deal with mixed numbers. To work with them, they all need to be converted to ordinary fractions. How to do it? You can catch a sixth grader and ask him. But not always a sixth grader will be at hand ... We will have to do it ourselves. It is not difficult. Multiply the denominator of the fractional part by the integer part and add the numerator of the fractional part. This will be the numerator of a common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but it's actually quite simple. Let's see an example.
Let in the problem you saw with horror the number:
Calmly, without panic, we understand. The whole part is 1. One. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of an ordinary fraction. That's all. It looks even simpler in mathematical notation:
Clearly? Then secure your success! Convert to common fractions. You should get 10/7, 7/2, 23/10 and 21/4.
The reverse operation - converting an improper fraction into a mixed number - is rarely required in high school. Well, if... And if you - not in high school - you can look into the special Section 555. In the same place, by the way, you will learn about improper fractions.
Well, almost everything. You remembered the types of fractions and understood How convert them from one type to another. The question remains: For what do it? Where and when to apply this deep knowledge?
I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed into a bunch, we translate everything into ordinary fractions. It can always be done. Well, if something like 0.8 + 0.3 is written, then we think so, without any translation. Why do we need extra work? We choose the solution that is convenient us !
If the task is full of decimal fractions, but um ... some kind of evil ones, go to ordinary ones, try it! Look, everything will be fine. For example, you have to square the number 0.125. Not so easy if you have not lost the habit of the calculator! Not only do you need to multiply the numbers in a column, but also think about where to insert the comma! It certainly doesn't work in my mind! And if you go to an ordinary fraction?
0.125 = 125/1000. We reduce by 5 (this is for starters). We get 25/200. Once again on 5. We get 5/40. Oh, it's shrinking! Back to 5! We get 1/8. Easily square (in your mind!) and get 1/64. All!
Let's summarize this lesson.
1. There are three types of fractions. Ordinary, decimal and mixed numbers.
2. Decimals and mixed numbers Always can be converted to common fractions. Reverse Translation not always available.
3. The choice of the type of fractions for working with the task depends on this very task. If there are different types of fractions in one task, the most reliable thing is to switch to ordinary fractions.
Now you can practice. First, convert these decimal fractions to ordinary ones:
3,8; 0,75; 0,15; 1,4; 0,725; 0,012
You should get answers like this (in a mess!):
On this we will finish. In this lesson, we brushed up on the key points on fractions. It happens, however, that there is nothing special to refresh ...) If someone has completely forgotten, or has not mastered it yet ... Those can go to a special Section 555. All the basics are detailed there. Many suddenly understand everything are starting. And they solve fractions on the fly).
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