Before you start studying algebraic fractions We recommend that you remember how to work with ordinary fractions.
Any fraction that has a letter factor is called an algebraic fraction.
Examples algebraic fractions.
Like a common fraction, an algebraic fraction has a numerator (at the top) and a denominator (at the bottom).
Reducing an algebraic fraction
Algebraic fractions can be reduced. When reducing, use the rules for reducing ordinary fractions.
We remind you that when reducing a common fraction, we divided both the numerator and the denominator by the same number.
An algebraic fraction can be reduced in the same way, but only the numerator and denominator are divided by the same polynomial.
Let's consider example of reducing an algebraic fraction.
Let us determine the smallest power in which the monomial “a” appears. The smallest power for the monomial "a" is in the denominator - this is the second power.
Let's divide both the numerator and the denominator by “a 2”. When dividing monomials, we use the property of quotient powers.
We remind you that any letter or number to the zero power is a unit.
There is no need to write down in detail each time what the algebraic fraction was reduced by. It is enough to keep in mind the degree by which the reduction was made and write down only the result.
A short notation for reducing an algebraic fraction is as follows.
Only identical letter factors can be abbreviated.
Can't shorten
Can be shortened
Other examples of reducing algebraic fractions.
How to reduce a fraction with polynomials
Let's look at another example of an algebraic fraction. You need to reduce an algebraic fraction that has a polynomial in its numerator.
You can reduce a polynomial in brackets only with exactly the same polynomial in brackets!
No way you can't shorten a part polynomial inside brackets!
Wrong
Determining where a polynomial ends is very simple. Between polynomials there can only be a multiplication sign. The entire polynomial is inside the parentheses.
After we have defined the polynomials of an algebraic fraction, we can cancel the polynomial “(m − n)” in the numerator with the polynomial “(m − n)” in the denominator.
Examples of reducing algebraic fractions with polynomials.
Subtracting a common factor when reducing fractions
In order for identical polynomials to appear in algebraic fractions, it is sometimes necessary to remove common multiplier out of brackets.
In this form it is impossible to reduce an algebraic fraction, since the polynomial
"(3f + k)" can only be reduced with the polynomial "(3f + k)".
Therefore, in order to get “(3f + k)” in the numerator, we take out the common factor “5”.
Reducing fractions using abbreviated multiplication formulas
In other examples, reducing algebraic fractions requires
application of abbreviated multiplication formulas. 
In its original form, it is impossible to reduce an algebraic fraction, since there are no identical polynomials.
But if we apply the formula for the difference of squares for the polynomial “(a 2 − b 2)”, then identical polynomials will appear.
More examples of reducing algebraic fractions using abbreviated multiplication formulas.
Reduction of algebraic (rational) fractions is based on their basic property: if the numerator and denominator of a fraction are divided by the same non-zero polynomial, then an equal fraction is obtained.
You can only reduce multipliers!
Members of polynomials cannot be abbreviated!
To reduce an algebraic fraction, the polynomials in the numerator and denominator must first be factorized.
Let's look at examples of reducing fractions.
The numerator and denominator of the fraction contain monomials. They represent work(numbers, variables and their powers), multipliers we can reduce.
We reduce the numbers by their greatest common divisor, that is, by greatest number, by which each of these numbers is divided. For 24 and 36 this is 12. After reduction from 24 there remains 2, from 36 - 3.
We reduce the degrees by degree c the lowest rate. To reduce a fraction means to divide the numerator and denominator by the same divisor, and when dividing powers, we subtract the exponents.
a² and a⁷ are reduced to a². In this case, one remains in the numerator of a² (we write 1 only in the case when, after reduction, there are no other factors left. From 24, 2 remains, so we do not write 1 remaining from a²). From a⁷, after reduction, a⁵ remains.
b and b are reduced by b; the resulting units are not written.
c³º and c⁵ are shortened to c⁵. From c³º what remains is c²⁵, from c⁵ is one (we don’t write it). Thus,
The numerator and denominator of this algebraic fraction are polynomials. You cannot cancel terms of polynomials! (you cannot reduce, for example, 8x² and 2x!). To reduce this fraction, you need to factor the polynomials. The numerator has a common factor of 4x. Let's take it out of brackets:
Both the numerator and denominator have the same factor (2x-3). We reduce the fraction by this factor. In the numerator we got 4x, in the denominator - 1. According to 1 property of algebraic fractions, the fraction is equal to 4x.
You can only reduce multipliers (reduce given fraction on 25x² it is impossible!). Therefore, the polynomials in the numerator and denominator of the fraction must be factorized.
In the numerator - perfect square sums, the denominator is the difference of squares. After decomposition using abbreviated multiplication formulas, we obtain:
We reduce the fraction by (5x+1) (to do this, cross out the two in the numerator as an exponent, leaving (5x+1)² (5x+1)):
The numerator has a common factor of 2, let's take it out of brackets. The denominator is the formula for the difference of cubes:
As a result of the expansion, the numerator and denominator received the same factor (9+3a+a²). We reduce the fraction by it:
The polynomial in the numerator consists of 4 terms. We group the first term with the second, the third with the fourth, and remove the common factor x² from the first brackets. We decompose the denominator using the sum of cubes formula:
In the numerator, let’s take the common factor (x+2) out of brackets:
Reduce the fraction by (x+2):
We can only reduce multipliers! To reduce this fraction, you need to factor the polynomials in the numerator and denominator. In the numerator the common factor is a³, in the denominator - a⁵. Let's take them out of brackets:
Factors - powers with the same base a³ and a⁵ - are reduced by a³. From a³ remains 1, we do not write it, from a⁵ remains a². In the numerator, the expression in parentheses can be expanded as a difference of squares:
We reduce the fraction by a common divisor (1+a):
How to reduce fractions of the form
in which the expressions in the numerator and denominator differ only in signs?
We will look at examples of reducing such fractions next time.
2 comments
Very good site, I use it every day and it helps.
Before I came across this site, I didn’t know how to solve much in algebra and geometry, but thanks to this site my grades a 3 went up by 4-5.
Now I can safely take the OGE, and they are afraid that I won’t pass it!
Learn and you will succeed!
Vitya, I wish you success in your studies and high results on exams!
www.algebraclass.ru
Reducing algebraic fractions rule
Reducing algebraic fractions
A new concept in mathematics rarely arises “out of nothing”, “on empty space" It appears when it is felt objective necessity. This is exactly how they appeared in mathematics negative numbers, this is how ordinary and decimal numbers appeared algebraic fraction.
We have the prerequisites for introducing the new concept of “algebraic fraction”. Let's return to § 12. While discussing the division of a monomial by a monomial there, we looked at a number of examples. Let's highlight two of them.
1. Divide the monomial 36a 3 b 5 by the monomial 4ab 2 (see example 1c) from §12).
This is how we solved it. Instead of writing 36a 3 b 5: 4ab 2, a fraction line was used:
This allowed us to also use the fraction line instead of 36: 4, a 3: a, b 5: b 2, which made the solution to the example more clear:
2. Divide the monomial 4x 3 by the monomial 2xy (see example 1 d) from § 12). Following the same pattern, we got:
In § 12 we noted that the monomial 4x 3 could not be divided by the monomial 2xy so as to obtain monomial. But mathematical models real situations can contain the division operation of any monomials, not necessarily such that one is divisible by the other. Anticipating this, mathematicians introduced a new concept - the concept of algebraic fraction. In particular, an algebraic fraction. Now let's return to § 18. Discussing there the operation of dividing a polynomial by a monomial, we noted that it is not always feasible. So, in example 2 from § 18 we were talking about dividing the binomial 6x 3 - 24x 2 by the monomial 6x 2. This operation turned out to be feasible and as a result we obtained the binomial x - 4. This means 
In other words, algebraic expression managed to replace more simple expression- polynomial x - 4.
At the same time, in example 3 from § 18, it was not possible to divide the polynomial 8a 3 + Ba 2b - b by 2a 2, i.e., the expression could not be replaced with a simpler expression, it had to be left in the form of an algebraic fraction.
As for the operation of dividing a polynomial by polynomial, then we actually didn’t say anything about it. The only thing we can say now is that one polynomial can be divided by another if that other polynomial is one of the factors in the factorization of the first polynomial.
For example, x 3 - 1 = (x - 1) (x 2 + x + 1). This means that x 3 - 1 can be divided by x 2 + x + 1, the result is x - 1; x 3 - 1 can be divided by x - 1,
it turns out x 2 + x + 1.
polynomials P and Q. In this case, use the notation
where P is the numerator, Q is the denominator of the algebraic fraction.
Examples of algebraic fractions:
Sometimes an algebraic fraction can be replaced by a polynomial. For example, as we established earlier,
(the polynomial 6x 3 - 24x 2 was divided by 6x 2, and in the quotient we get x - 4); we also noted that
But this happens relatively rarely.
However, you have already encountered a similar situation - when studying ordinary fractions. For example, a fraction - can be replaced by the integer 4, and a fraction - by the integer 5. However, a fraction - cannot be replaced by an integer, although this fraction can be reduced by dividing the numerator and denominator by the number 8 - the common factor of the numerator and denominator:
In the same way, you can reduce algebraic fractions by simultaneously dividing the numerator and denominator of the fraction by their common multiplier. And to do this, you need to factor both the numerator and the denominator of the fraction. This is where we need everything that we have discussed for so long in this chapter.
Example. Reduce an algebraic fraction:
Solution, a) Find the common factor for the monomials
12x 3 y 4 and 8x 2 y 5 as we did in § 20. We get 4x 2 y 4. Then 12x 3 y 4 = 4x 2 y 4 3x; 8x 2 y 5 = 4x 2 y 4 2y.
Means,
Numerator and denominator a given algebraic fraction was reduced by a common factor 4x 2 y 4.
The solution to this example can be written differently:
b) To reduce a fraction, factor its numerator and denominator. We get:
(the fraction was reduced by a common factor a + b).
Now return to remark 2 from § 1. You see, we were finally able to fulfill the promise made there.
c) We have:
(reduced the fraction by the common factor of the numerator and denominator, i.e. by x (x - y))
So, in order to reduce an algebraic fraction, you must first factorize its numerator and denominator. So your success in this new activity (reducing algebraic fractions) largely depends on how well you have mastered the material in the previous paragraphs of this chapter.
A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions
If you have any corrections or suggestions for this lesson, write to us.
If you want to see other adjustments and suggestions for lessons, look here - Educational Forum.
Reducing algebraic fractions: rules, examples.
We continue to study the topic of converting algebraic fractions. In this article we will go into detail about reducing 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. Below we present a rule that allows this transformation to be carried out. Finally, let's look at solutions typical examples, which will allow you to understand all the intricacies of the process.
Page navigation.
What does it mean to reduce an algebraic fraction?
While studying common fractions, we talked about their reduction. We call a reduction of a common fraction the division of its numerator and denominator by a common factor. For example, the common fraction 30/54 can be reduced by 6 (that is, its numerator and denominator divided by 6), which leads us to the fraction 5/9.
By reducing an algebraic fraction we mean a similar action. Reduce an algebraic fraction- this means dividing 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 a fraction 
. It is also possible to perform a contraction to the variable x, resulting in the expression 
. The original algebraic fraction can be reduced to the monomial 3 x, as well as to 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 more simple type, V best case scenario– irreducible fraction.
Can any algebraic fraction be reduced?
We know that ordinary fractions are divided into reducible and irreducible fractions. Irreducible fractions do not have common factors in the numerator and denominator other than one, and therefore cannot be reduced.
Algebraic fractions may or may not have common factors in the numerator and denominator. If there are common factors, it is possible to reduce an algebraic fraction. If there are no common factors, then simplifying an algebraic fraction by reducing it is impossible.
In general, according to appearance algebraic fraction, it is quite difficult to determine whether it can be reduced. Of course, 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 3. It is also easy to notice that an algebraic fraction can be reduced by x, by y, or directly 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, it is possible to reduce a fraction by x−1, but this common factor is not clearly present in the notation. And an algebraic fraction 
it is impossible to reduce, since its numerator and denominator do not have common factors.
In general, the question of the reducibility of an algebraic fraction is very difficult. And sometimes it is easier to solve a problem by working with an algebraic fraction in its original form than to find out whether this fraction can be reduced first. But there are still transformations that in some cases make it possible, 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.
Rule for reducing algebraic fractions
The information from the previous paragraphs allows you to naturally perceive the following rule for reducing algebraic fractions, which consists of two steps:
- first, the common factors of the numerator and denominator of the original fraction are found;
- if there are any, then a reduction is made by these factors.
The indicated steps of the announced rule need clarification.
Most convenient way finding common ones consists in factoring 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 common factors are found, then in the second step they are reduced. The result is a new fraction of a simpler form.
The rule for reducing algebraic fractions is based on the basic 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 the reduction is performed - the transition to the fraction.
Let's move on to solving examples using of this rule. On them we will analyze all the possible nuances that arise when factoring the numerator and denominator of an algebraic fraction into factors and subsequent reduction.
Typical examples
First, we need to talk about reducing algebraic fractions whose numerator and denominator are the same. Such fractions are identically equal to one on the entire ODZ of the variables included in it, for example, 
etc.
Now it doesn’t hurt to remember how to reduce ordinary fractions - after all, they are a special case of algebraic fractions. The natural numbers in the numerator and denominator of a common fraction are broken down into prime factors, after which the common factors are canceled (if any). For example, 
. Product of identical prime factors can be written in the form of powers, and when abbreviating, use the property of dividing powers with on the same grounds. 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.
Absolutely similar principles are used to reduce algebraic fractions, the numerator and denominator of which contain monomials with integer coefficients.
Cancel an algebraic fraction 
.
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 in the form of an expression with powers: 
.
Regarding the reduction of algebraic fractions having fractional numerical odds in the numerator and denominator, then you can do two things: either separately divide these fractional odds, or first get rid of 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 basic property of an algebraic fraction. Let's understand this with an example.
Perform fraction reduction.
You can reduce the fraction as follows: 
.
Or you could first get rid of fractional coefficients by multiplying the numerator and denominator by the least common multiple of the denominators of these coefficients, that is, by LCM(5, 10)=10. In this case we have 
.
.
We can move on to algebraic fractions general view, 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 verify its absence, you need to factor the numerator and denominator of an algebraic fraction.
Reduce a rational fraction 
.
To do this, factor the polynomials in the numerator and denominator. Let's start by putting it out of brackets: . Obviously, the expressions in parentheses can be converted using abbreviated multiplication formulas: 
. Now it is clearly seen that it is possible to reduce the fraction by a common factor b 2 ·(a+7) . Let's do it 
.
Quick Solution without explanation, they are usually written as a chain of equalities: 
.
Sometimes common factors can be hidden by numeric odds. Therefore, when reducing rational fractions It is advisable to put numerical factors at higher powers of the numerator and denominator out of brackets.
Reduce the fraction 
, if possible.
At first glance, the numerator and denominator do not have a common factor. But still, let's try to perform some transformations. First, you can take out the factor x in the numerator: 
.
Now there is some similarity between the expression in brackets and the expression in the denominator due to x 2 ·y. Let us take out the numerical coefficients of the higher powers of these polynomials: 
After the transformations have been made, a common factor is visible, by which we carry out the reduction. We have 
.
Concluding the conversation about reducing rational fractions, we note that success largely depends on the ability to factor polynomials.
www.cleverstudents.ru
Mathematics
Navigation bar
Reducing algebraic fractions
Based on the above property, we can simplify algebraic fractions in the same way as we do with arithmetic fractions, cutting them down.
Reducing fractions involves dividing the numerator and denominator of the fraction by the same number.
If the algebraic fraction is single-term, then the numerator and denominator are represented as a product of several factors, and it is immediately clear which same numbers you can separate them:
We can write the same fraction in more detail: . We see that we can sequentially divide both the numerator and the denominator 4 times by a, that is, ultimately divide each of them by a 4. That's why ; also, etc. So, if the numerator and denominator have factors of different powers of the same letter, then this fraction can be reduced by a lower power of this letter.
If the fraction is polynomial, then you first have to factor these polynomials, if possible, into factors, and then it will be possible to see into what identical factors both the numerator and the denominator can be divided.
…. the numerator is easily factored “according to the formula” - it is the square of the difference between two numbers, namely (x – 3) 2. The denominator does not fit the formulas and it will have to be expanded using the technique used for quadratic trinomial: let’s find 2 numbers so that their sum is –1 and their product = –6, – these numbers are –3 and + 2; then x 2 – x – 6 = x 2 – 3x + 2x – 6 = x (x – 3) + 2 (x – 3) = (x – 3) (x + 2).
Popular:
- Brief rules chess games CHESS BOARD AND NOTATION Chess is a game for two. One player (White) uses pieces white, and the second player (Black) usually plays with black pieces. The board is divided into 64 small […]
- Simplifying Expressions The properties of addition, subtraction, multiplication, and division are useful because they allow you to transform sums and products into convenient expressions for calculations. Let's learn how to use these properties to simplify [...]
- Inertia rules Dynamics is a branch of mechanics that studies the movement of bodies under the influence of forces applied to them. Biomechanics also considers the interaction between the human body and external environment, between the links of the body, [...]
- The letters e (е), o after the sibilants at the root of the word. Rules and examples We will choose the spelling of the letters “e” (ё) or “o” after hissing words at the root using the corresponding rule of Russian spelling. Let's see how […]
- Mechanical and electromagnetic vibrations 4. Oscillations and waves 1. Harmonic vibrations the values of s are described by the equation s = 0.02 cos (6πt + π/3), m. Determine: 1) the amplitude of oscillations; 2) cyclic frequency; 3) frequency […]
- Ostwald dilution law 4.6 Ostwald dilution law Degree of dissociation (αdis) and dissociation constant (Kdis) weak electrolyte quantitatively related to each other. Let us derive the equation for this connection using the example of a weak […]
- Wording and content of the order of the Ministry of Defense of the Russian Federation No. 365 of 2002 This order contains information on the right to additional days of vacation depending on various conditions and aspects of service. This order is silent [...]
- Impose disciplinary action have the right Chapter 3. DISCIPLINARY SANCTIONS The rights of commanders (chiefs) to impose disciplinary sanctions on their subordinate warrant officers and midshipmen 63. The platoon (group) commander and […]
To understand how to reduce fractions, let's first look at an example.
To reduce a fraction means to divide the numerator and denominator by the same thing. Both 360 and 420 end in a digit, so we can reduce this fraction by 2. In the new fraction, both 180 and 210 are also divisible by 2, so we reduce this fraction by 2. In the numbers 90 and 105, the sum of the digits is divisible by 3, so both these numbers are divisible by 3, we reduce the fraction by 3. In the new fraction, 30 and 35 end in 0 and 5, which means both numbers are divisible by 5, so we reduce the fraction by 5. The resulting fraction of six-sevenths is irreducible. This is the final answer.
We can arrive at the same answer in a different way.
Both 360 and 420 end in zero, which means they are divisible by 10. We reduce the fraction by 10. In the new fraction, both the numerator 36 and the denominator 42 are divided by 2. We reduce the fraction by 2. In the next fraction, both the numerator 18 and the denominator 21 are divided by 3, which means we reduce the fraction by 3. We came to the result - six sevenths.
And one more solution.
Next time we'll look at examples of reducing fractions.
This article continues the topic of converting algebraic fractions: consider such an action as reducing algebraic fractions. Let's define the term itself, formulate a reduction rule and analyze practical examples.
Yandex.RTB R-A-339285-1
The meaning of reducing an algebraic fraction
In materials about common fractions, we looked at its reduction. We defined reducing a fraction as dividing its numerator and denominator by a common factor.
Reducing an algebraic fraction is a similar operation.
Definition 1
Reducing an algebraic fraction is the division of its numerator and denominator by a common factor. In this case, in contrast to the reduction of an ordinary fraction (the common denominator can only be a number), 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, the algebraic fraction 3 x 2 + 6 x y 6 x 3 y + 12 x 2 y 2 can be reduced by the number 3, resulting in: x 2 + 2 x y 6 x 3 · y + 12 · x 2 · y 2 . We can reduce the same fraction by the variable x, and this will give us the expression 3 x + 6 y 6 x 2 y + 12 x y 2. Also given fraction can be reduced by a monomial 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 reduction of an algebraic fraction is a fraction of a simpler form, at best an irreducible fraction.
Are all algebraic fractions subject to reduction?
Again, from materials on ordinary fractions, we know that there are reducible and irreducible fractions. Irreducible fractions are fractions that do not have common numerator and denominator factors other than 1.
It’s the same with algebraic fractions: they may have common factors in the numerator and denominator, or they may not. The presence of common factors allows you to simplify the original fraction through reduction. When there are no common factors, it is impossible to optimize a given fraction using the reduction method.
IN general cases By given type It is quite difficult for a fraction to understand whether it can be reduced. Of course, in some cases the presence of a common factor between the numerator and denominator is obvious. For example, in the algebraic fraction 3 x 2 3 y it is quite clear that the common factor is the number 3.
In the fraction - x · y 5 · x · y · z 3 we also immediately understand that it can be reduced by x, or y, or x · y. And yet, much more often there are examples of algebraic fractions, when the common factor of the numerator and denominator is not so easy to see, and even more often, it is simply absent.
For example, we can reduce the fraction x 3 - 1 x 2 - 1 by x - 1, while the specified common factor is not present in the entry. But the fraction x 3 - x 2 + x - 1 x 3 + x 2 + 4 · x + 4 cannot be reduced, since the numerator and denominator do not have a common factor.
Thus, the question of determining the reducibility of an algebraic fraction is not so simple, and it is often easier to work with a fraction of a given form than to try to find out whether it is reducible. In this case, such transformations take place that in particular cases make it possible to determine the common factor of the numerator and denominator or to draw a conclusion about the irreducibility of a fraction. We will examine this issue in detail in the next paragraph of the article.
Rule for reducing algebraic fractions
Rule for reducing algebraic fractions consists of two sequential actions:
- finding common factors of the numerator and denominator;
- if any are found, the action of reducing the fraction is carried out directly.
The most convenient method of finding common denominators is to factor the polynomials present in the numerator and denominator of a given algebraic fraction. This allows you to immediately clearly see the presence or absence of common factors.
The very action of reducing an algebraic fraction is based on the main property of an algebraic fraction, expressed by the equality undefined, where a, b, c are some polynomials, and b and c are non-zero. The first step is to reduce the fraction to the form a · c b · c, in which we immediately notice the common factor c. The second step is to perform a reduction, i.e. transition to a fraction of the form a b .
Typical examples
Despite some obviousness, let us clarify about special case when the numerator and denominator of an algebraic fraction are equal. Similar fractions are identically equal to 1 on the entire ODZ of the variables of this fraction:
5 5 = 1 ; - 2 3 - 2 3 = 1 ; x x = 1 ; - 3, 2 x 3 - 3, 2 x 3 = 1; 1 2 · x - x 2 · y 1 2 · x - x 2 · y ;
Since ordinary fractions are a special case of algebraic fractions, let us recall how they are reduced. The natural numbers written in the numerator and denominator are factored into prime factors, then the common factors are canceled (if any).
For example, 24 1260 = 2 2 2 3 2 2 3 3 5 7 = 2 3 5 7 = 2 105
The product of simple identical factors can be written as powers, and in the process of reducing a fraction, use the property of dividing powers with identical bases. Then the above solution would be:
24 1260 = 2 3 3 2 2 3 2 5 7 = 2 3 - 2 3 2 - 1 5 7 = 2 105
(numerator and denominator divided by a common factor 2 2 3). Or for clarity, based on the properties of multiplication and division, we give the solution the following form:
24 1260 = 2 3 3 2 2 3 2 5 7 = 2 3 2 2 3 3 2 1 5 7 = 2 1 1 3 1 35 = 2 105
By analogy, the reduction of algebraic fractions is carried out, in which the numerator and denominator have monomials 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 needs to be reduced.
Solution
It is possible to write the numerator and denominator of a given fraction as a product of simple factors and variables, and then carry out the reduction:
27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z = - 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 = = - 3 · 3 · a · a · a 2 · c · c · c · c · c · c = - 9 a 3 2 c 6
However, more in a rational way the solution will be written in the form of an expression with degrees:
27 · a 5 · b 2 · c · z 6 · a 2 · b 2 · c 7 · z = - 3 3 · a 5 · b 2 · c · z 2 · 3 · a 2 · b 2 · c 7 · z = - 3 3 2 · 3 · a 5 a 2 · b 2 b 2 · c c 7 · z z = = - 3 3 - 1 2 · a 5 - 2 1 · 1 · 1 c 7 - 1 · 1 = · - 3 2 · a 3 2 · c 6 = · - 9 · a 3 2 · c 6 .
Answer:- 27 a 5 b 2 c z 6 a 2 b 2 c 7 z = - 9 a 3 2 c 6
When the numerator and denominator of an algebraic fraction have fractional numerical coefficients, two ways are possible further actions: either separately divide these fractional coefficients, or first get rid of the fractional coefficients by multiplying the numerator and denominator by a certain natural number. The last transformation is carried out due to the basic property of an algebraic fraction (you can read about it in the article “Reducing an algebraic fraction to a new denominator”).
Example 2
The given fraction is 2 5 x 0, 3 x 3. It needs to be reduced.
Solution
It is possible to reduce the fraction this way:
2 5 x 0, 3 x 3 = 2 5 3 10 x x 3 = 4 3 1 x 2 = 4 3 x 2
Let's try to solve the problem differently, having first gotten rid of fractional coefficients - multiply the numerator and denominator by the least common multiple of the denominators of these coefficients, i.e. on LCM (5, 10) = 10. Then we get:
2 5 x 0, 3 x 3 = 10 2 5 x 10 0, 3 x 3 = 4 x 3 x 3 = 4 3 x 2.
Answer: 2 5 x 0, 3 x 3 = 4 3 x 2
When we reduce general algebraic fractions, in which the numerators and denominators can be either monomials or polynomials, there can be a problem where the common factor is not always immediately visible. Or moreover, it simply does not exist. Then, to determine the common factor or record the fact of its absence, the numerator and denominator of the algebraic fraction are factored.
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 needs to be reduced.
Solution
Let us factor the polynomials in the numerator and denominator. Let's put it out of brackets:
2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49)
We see that the expression in parentheses can be converted using abbreviated multiplication formulas:
2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49) = 2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7)
It is clearly seen that it is possible to reduce a fraction by a common factor b 2 (a + 7). Let's make a reduction:
2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7) = 2 (a + 7) b (a - 7) = 2 a + 14 a b - 7 b
Let us write a short solution without explanation as a chain of equalities:
2 a 2 b 2 + 28 a b 2 + 98 b 2 a 2 b 3 - 49 b 3 = 2 b 2 (a 2 + 14 a + 49) b 3 (a 2 - 49) = = 2 b 2 (a + 7) 2 b 3 (a - 7) (a + 7) = 2 (a + 7) b (a - 7) = 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 = 2 a + 14 a b - 7 b.
It happens that common factors are hidden by numerical coefficients. Then, when reducing fractions, it is optimal to put the numerical factors at higher powers of the numerator and denominator out of brackets.
Example 4
Given the algebraic fraction 1 5 · x - 2 7 · x 3 · y 5 · x 2 · y - 3 1 2 . It is necessary to reduce it if possible.
Solution
At first glance, the numerator and denominator do not exist common denominator. However, let's try to convert the given fraction. Let's take out the factor x in the numerator:
1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 = x 1 5 - 2 7 x 2 y 5 x 2 y - 3 1 2
Now you can see some similarity between the expression in brackets and the expression in the denominator due to x 2 y . Let us take out the numerical coefficients of the higher powers of these polynomials:
x 1 5 - 2 7 x 2 y 5 x 2 y - 3 1 2 = x - 2 7 - 7 2 1 5 + x 2 y 5 x 2 y - 1 5 3 1 2 = = - 2 7 x - 7 10 + x 2 y 5 x 2 y - 7 10
Now the common factor becomes visible, we carry out the reduction:
2 7 x - 7 10 + x 2 y 5 x 2 y - 7 10 = - 2 7 x 5 = - 2 35 x
Answer: 1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 = - 2 35 x .
Let us emphasize that the skill of reducing rational fractions depends on the ability to factor polynomials.
If you notice an error in the text, please highlight it and press Ctrl+Enter
Online calculator performs reduction of algebraic fractions in accordance with the rule of reducing fractions: replacing the original fraction equal fraction, but with a smaller numerator and denominator, i.e. Simultaneously dividing the numerator and denominator of a fraction by their common greatest common factor (GCD). The calculator also displays a detailed solution that will help you understand the sequence of the reduction.
Given:
Solution:
Performing fraction reduction
checking the possibility of performing algebraic fraction reduction
1) Determination of the greatest common divisor (GCD) of the numerator and denominator of a fraction
determination of the largest common divisor(GCD) of the numerator and denominator of an algebraic fraction
2) Reducing the numerator and denominator of a fraction
reducing the numerator and denominator of an algebraic fraction
3) Selecting the whole part of a fraction
separating the whole part of an algebraic fraction
4) Converting an algebraic fraction to a decimal fraction
converting an algebraic fraction to decimal
Help for website development of the project
Dear Site Visitor.
If you were unable to find what you were looking for, be sure to write about it in the comments, what is currently missing on the site. This will help us understand in which direction we need to move further, and other visitors will soon be able to receive the necessary material.
If the site turned out to be useful to you, donate the site to the project only 2 ₽ and we will know that we are moving in the right direction.
Thank you for stopping by!
I. Procedure for reducing an algebraic fraction using an online calculator:
- To reduce an algebraic fraction, enter the values of the numerator and denominator of the fraction in the appropriate fields. If the fraction is mixed, then also fill in the field corresponding to the whole part of the fraction. If the fraction is simple, then leave the whole part field blank.
- To set negative fraction, put a minus sign on the whole part of the fraction.
- Depending on the specified algebraic fraction, the following sequence of actions is automatically performed:
- determining the greatest common divisor (GCD) of the numerator and denominator of a fraction;
- reducing the numerator and denominator of a fraction by gcd;
- highlighting the whole part of a fraction, if the numerator of the final fraction is greater than the denominator.
- converting the final algebraic fraction to a decimal fraction rounded to the nearest hundredth.
II. For reference:
A fraction is a number consisting of one or more parts (fractions) of a unit. Common fraction(simple fraction) is written as two numbers (the numerator of the fraction and the denominator of the fraction) separated by a horizontal bar (the fraction bar) indicating the division sign. The numerator of a fraction is the number above the fraction line. The numerator shows how many shares were taken from the whole. The denominator of a fraction is the number below the fraction line. The denominator shows how many equal parts the whole is divided into. A simple fraction is a fraction that does not have a whole part. A simple fraction can be proper or improper. proper fraction - a fraction whose numerator is less than the denominator, so a proper fraction is always less than one. Example of proper fractions: 8/7, 11/19, 16/17. improper fraction - a fraction whose numerator is greater than or equal to the denominator, so an improper fraction is always greater than or equal to one. Example of improper fractions: 7/6, 8/7, 13/13. mixed fraction is a number that contains a whole number and a proper fraction, and denotes the sum of that whole number and the proper fraction. Any mixed fraction can be converted to an improper fraction simple fraction. Example mixed fractions: 1¼, 2½, 4¾.
III. Note:
- Source data block highlighted yellow , intermediate calculation block allocated blue , the solution block is highlighted in green.
- To add, subtract, multiply and divide ordinary or mixed fractions, use the online fraction calculator with detailed solution.
