It 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 will be 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 the largest number by which each of these numbers is divided. For 24 and 36 this is 12. After reduction, 2 remains from 24, and 3 from 36.
We reduce the degrees by the degree with the lowest index. To reduce a fraction means to divide the numerator and denominator by the same divisor, and 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. What remains from c³º 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 . 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 factors (you cannot reduce this fraction by 25x²!). Therefore, the polynomials in the numerator and denominator of the fraction must be factorized.
The numerator is the complete square of the sum, 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. 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):
Division and the numerator and denominator of the fraction on their common divisor, different from one, is called reducing a fraction.
To reduce a common fraction, you need to divide its numerator and denominator by the same natural number.
This number is the greatest common divisor of the numerator and denominator of the given fraction.
The following are possible decision recording forms Examples for reducing common fractions.
The student has the right to choose any form of recording.
Examples. Simplify fractions.
Reduce the fraction by 3 (divide the numerator by 3;
divide the denominator by 3).
Reduce the fraction by 7.
We perform the indicated actions in the numerator and denominator of the fraction.
The resulting fraction is reduced by 5.
Let's reduce this fraction 4)
on 5·7³- the greatest common divisor (GCD) of the numerator and denominator, which consists of the common factors of the numerator and denominator, taken to the power with the smallest exponent.
Let's factor the numerator and denominator of this fraction into prime factors.
We get: 756=2²·3³·7 And 1176=2³·3·7².
Determine the GCD (greatest common divisor) of the numerator and denominator of the fraction 5) .
This is the product of common factors taken with the lowest exponents.
gcd(756, 1176)= 2²·3·7.
We divide the numerator and denominator of this fraction by their gcd, i.e. by 2²·3·7 we get an irreducible fraction 9/14
.
Or it was possible to write the decomposition of the numerator and denominator in the form of a product of prime factors, without using the concept of power, and then reduce the fraction by crossing out the same factors in the numerator and denominator. When there are no identical factors left, we multiply the remaining factors separately in the numerator and separately in the denominator and write out the resulting fraction 9/14 .
And finally, it was possible to reduce this fraction 5) gradually, applying signs of dividing numbers to both the numerator and denominator of the fraction. Let's think like this: numbers 756 And 1176 end in an even number, which means both are divisible by 2 . We reduce the fraction by 2 . The numerator and denominator of the new fraction are numbers 378 And 588 also divided into 2 . We reduce the fraction by 2 . We notice that the number 294 - even, and 189 is odd, and reduction by 2 is no longer possible. Let's check the divisibility of numbers 189 And 294 on 3 .
(1+8+9)=18 is divisible by 3 and (2+9+4)=15 is divisible by 3, hence the numbers themselves 189 And 294 are divided into 3 . We reduce the fraction by 3 . Further, 63 is divisible by 3 and 98 - No. Let's look at other prime factors. Both numbers are divisible by 7 . We reduce the fraction by 7 and we get the irreducible fraction 9/14 .
Goals:
1. Educational- consolidate the acquired knowledge and skills of reducing algebraic fractions when solving more complex exercises, using factorization of a polynomial in different ways, and develop the ability to reduce algebraic fractions. Repeat abbreviated multiplication formulas: (a+b)2=a2+2ab+b2,
(a-b) 2 =a 2 -2ab+b 2,a 2 -b 2 =(a+b)(a-b), method of grouping, placing the common factor out of brackets.
2. Developmental – development of logical thinking for conscious perception of educational material, attention, activity of students in the lesson.
3. Educating - education of cognitive activity, formation of personal qualities: accuracy and clarity of verbal expression of thoughts; concentration and attention; perseverance and responsibility, positive motivation to study the subject, accuracy, conscientiousness and a sense of responsibility.
Tasks:
1. Reinforce the material studied by changing types of work on this topic “Algebraic fraction. Reducing fractions."
2. Develop skills and abilities in reducing algebraic fractions using different methods of factoring the numerator and denominator, develop logical thinking, correct and competent mathematical speech, develop independence and confidence in one’s knowledge and skills when performing different types of work.
3. To cultivate interest in mathematics by introducing different types of consolidation of material: oral work, work with a textbook, work at the blackboard, mathematical dictation, test, independent work, the game “Math Tournament”; stimulating and encouraging student activities.
Plan:
I. Organizing time.
II .
Oral work.
III. Mathematical dictation.
IV.
1.Work according to the textbook and at the blackboard.
2. Work in groups using cards - the game “Math Tournament”.
3. Independent work on levels (A, B, C).
V. Bottom line.
1. Test (mutual verification).
VI. Homework.
During the classes:
I. Organizational moment.
The emotional mood and readiness of the teacher and students for the lesson. Students set goals and objectives for this lesson, based on the teacher’s guiding questions, determine the topic of the lesson.
II. Oral work.
1. Reduce fractions:
2. Find the value of the algebraic fraction:
at c = 8, c = -13, c = 11.
Answer: 6; -1; 3.
3. Answer the questions:
1) What is the useful order to follow when factoring polynomials?
(When factoring polynomials, it is useful to follow the following order: a) put the common factor out of brackets, if there is one; b) try to factor the polynomial using abbreviated multiplication formulas; c) try to apply the grouping method if the previous methods did not lead to the goal).
2) What is the square of the sum?
(The square of the sum of two numbers is equal to the square of the first number plus twice the product of the first number and the second plus the square of the second number).
3) What is the square of the difference?
(The square of the difference of two numbers is equal to the square of the first number minus twice the product of the first number and the second plus the square of the second number).
4) What is the difference between the squares of two numbers?
(The difference between the squares of two numbers is equal to the product of the difference between these numbers and their sum).
5) What needs to be done when using the grouping method?
(To factor a polynomial using the grouping method, you need to: a) combine the members of the polynomial into groups that have a common factor in the form of a polynomial; b) take this common factor out of brackets).
6) To take the common factor out of brackets, you need......?
(Find this common factor; 2. put it out of brackets).
7) What methods do you know of factoring a polynomial?
(Putting the common factor out of brackets, grouping method, abbreviated multiplication formulas).
8) What is needed to reduce a fraction?
(To reduce a fraction, divide the numerator and denominator by their common factor.)
III. Mathematical dictation.
- Underline algebraic fractions:
Option I: 
Option II: 
- Is it possible to imagine the expression
Option I:
Option II:
as a polynomial? Can you imagine?
3. What letter values are acceptable for the expression:
Option I:
Option II:
(x-5)(x+7).
4. Write an algebraic fraction with a numerator
Option I:
3x2.
Option II:
5y.
and denominator
Option I:
x(x+3).
Option II:
y 2 (y+7).
and shorten it.
IV. Consolidation of the topic: “Algebraic fraction. Reducing fractions":
1.Work according to the textbook and at the blackboard.
Factor the numerator and denominator of the fraction and reduce it.
№441(1;3).
1. 
; 3. 
№442(1;3;5).
1. 
3. 
№443(1;3).
1. 3. 
1. 
3. 
1. 
3. 
2. Work in groups using cards - the game “Math Tournament”.
(Tasks for the game - “Appendix 1”.)
Consolidation and testing of skills in solving examples on this topic is carried out in the form of a tournament. The class is divided into groups and they are given tasks on cards (cards of different levels).
After a certain time, each student must write down the solution to his team’s assignments in a notebook and be able to explain them.
Consultations within the team are allowed (conducted by the captain).
Then the tournament begins: each team has the right to challenge others, but only once. For example, the captain of the first team calls students from the second team to participate in the tournament; The captain of the second team does the same, they go to the board, exchange cards and solve problems, etc.
3. Independent work on levels (A, B, C)
“Didactic material” L.I. Zvavich et al., p. 95, C-52. (the book is available to all students)
A
. №1:
I option-1) a, b; 2) a,c; 5) a.
II option-1) c, d; 2) b, d, 5) c.
B
. №2:
Option I - a.
Option II - b.
IN
. №3:
Option I - a.
Option II - b.
V. Bottom line.
1. Test (mutual verification).
(Tasks for the test - “Appendix 2”.)
(on cards for each student, according to options)
VI. Homework.
1) "D.M." page 95 No. 1. (3,4,6);
2) No. 447 (even);
3) §24, repeat § 19 - §23.
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. It is also possible to reduce a given fraction 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 of reducing 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 factors in the numerator and denominator 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, given the type of fraction it is quite difficult 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 the 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, a more rational way would be to write the solution as an expression with powers:
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 contain fractional numerical coefficients, there are two possible ways of further action: either divide these fractional coefficients separately, or first get rid of the fractional coefficients by multiplying the numerator and denominator by some 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's 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 have a 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 with an 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
determining the greatest 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 a decimal
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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 specify a negative fraction, place 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. A 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. A proper fraction is a fraction whose numerator is less than its denominator, so a proper fraction is always less than one. Example of proper fractions: 8/7, 11/19, 16/17. An improper fraction is a fraction in which the 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. Example of mixed fractions: 1¼, 2½, 4¾.
III. Note:
- The source data block is highlighted in yellow, the block of intermediate calculations is highlighted in blue, the solution block is highlighted in green.
- To add, subtract, multiply and divide common or mixed fractions, use the online fraction calculator with detailed solutions.