Identical factors are not reduced. Selecting the whole part of a fraction

This article continues the theme of transformation algebraic fractions: consider such an action as reducing algebraic fractions. Let's define the term itself, formulate a reduction rule and analyze practical examples.

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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 for 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 of reducing an algebraic fraction is a fraction greater than simple type, V best case scenario– 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 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 for 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 on the same grounds. 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 have fractional numerical coefficients, two ways are possible further actions: or separately divide these fractional odds, or first get rid of 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 algebraic fractions general view, in which the numerators and denominators can be either monomials or polynomials, there may be a problem when 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 contraction rational fractions depends on the ability to factor polynomials.

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In this article we will look in detail at how reducing fractions. First, let's discuss what is called reducing a fraction. After this, let's talk about reducing a reducible fraction to an irreducible form. Next we will obtain the rule for reducing fractions and, finally, consider examples of the application of this rule.

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What does it mean to reduce a fraction?

We know that ordinary fractions are divided into reducible and irreducible fractions. You can guess from the names that reducible fractions can be reduced, but irreducible fractions cannot.

What does it mean to reduce a fraction? Reduce fraction- this means dividing its numerator and denominator by their positive and different from unity. It is clear that as a result of reducing a fraction, a new fraction is obtained with a smaller numerator and denominator, and, due to the basic property of the fraction, the resulting fraction is equal to the original one.

For example, let's reduce the common fraction 8/24 by dividing its numerator and denominator by 2. In other words, let's reduce the fraction 8/24 by 2. Since 8:2=4 and 24:2=12, this reduction results in the fraction 4/12, which is equal to the original fraction 8/24 (see equal and unequal fractions). As a result, we have .

Reducing ordinary fractions to irreducible form

Typically, the ultimate goal of reducing a fraction is to obtain an irreducible fraction that is equal to the original reducible fraction. This goal can be achieved by reducing the original reducible fraction by its numerator and denominator. As a result of such a reduction, an irreducible fraction is always obtained. Indeed, a fraction is irreducible, since it is known that And - . Here we will say that the largest common divisor The numerator and denominator of the fraction are the largest number, by which this fraction can be reduced.

So, reducing a common fraction to an irreducible form consists of dividing the numerator and denominator of the original reducible fraction by their gcd.

Let's look at an example, for which we return to the fraction 8/24 and reduce it by the greatest common divisor of the numbers 8 and 24, which is equal to 8. Since 8:8=1 and 24:8=3, we come to the irreducible fraction 1/3. So, .

Note that the phrase “reduce a fraction” often means reducing the original fraction to its irreducible form. In other words, reducing a fraction very often refers to dividing the numerator and denominator by their greatest common factor (rather than by any common factor).

How to reduce a fraction? Rules and examples of reducing fractions

All that remains is to look at the rule for reducing fractions, which explains how to reduce a given fraction.

Rule for reducing fractions consists of two steps:

firstly, the gcd of the numerator and denominator of the fraction is found;
secondly, the numerator and denominator of the fraction are divided by their gcd, which gives an irreducible fraction equal to the original one.

Let's sort it out example of reducing a fraction according to the stated rule.

Example.

Reduce the fraction 182/195.

Solution.

Let's carry out both steps prescribed by the rule for reducing a fraction.

First we find GCD(182, 195) . It is most convenient to use the Euclid algorithm (see): 195=182·1+13, 182=13·14, that is, GCD(182, 195)=13.

Now we divide the numerator and denominator of the fraction 182/195 by 13, and we get the irreducible fraction 14/15, which is equal to the original fraction. This completes the reduction of the fraction.

Briefly, the solution can be written as follows: .

Answer:

This is where we can finish reducing fractions. But to complete the picture, let's look at two more ways to reduce fractions, which are usually used in easy cases.

Sometimes the numerator and denominator of the fraction being reduced is not difficult. Reducing a fraction in this case is very simple: you just need to remove all common factors from the numerator and denominator.

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

Let's look at the solution to the example.

Example.

Reduce the fraction 360/2 940.

Solution.

Let's factor the numerator and denominator into simple factors: 360=2·2·2·3·3·5 and 2,940=2·2·3·5·7·7. Thus, .

Now we get rid of the common factors in the numerator and denominator; for convenience, we simply cross them out: .

Finally, we multiply the remaining factors: , and the reduction of the fraction is completed.

Here short note solutions: .

Answer:

Let's consider another way to reduce a fraction, which consists of sequential reduction. Here, at each step, the fraction is reduced by some common divisor of the numerator and denominator, which is either obvious or easily determined using

When a student moves to high school, mathematics is divided into 2 subjects: algebra and geometry. There are more and more concepts, the tasks are more and more difficult. Some people have difficulty understanding fractions. Missed the first lesson on this topic, and voila. fractions? A question that will torment throughout my school life.

The concept of an algebraic fraction

Let's start with a definition. Under algebraic fraction refers to the expressions P/Q, where P is the numerator and Q is the denominator. A number may be hidden under the letter entry, numeric expression, numerical-letter expression.

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

As a rule, an integer is 1. The number in the denominator shows how many parts the unit is divided into. The numerator is needed to find out how many elements are taken. The fraction bar corresponds to the division sign. Recording allowed fractional expression as a mathematical operation "Division". In this case, the numerator is the dividend, the denominator is the divisor.

Basic rule of common fractions

When students pass this topic at school, they are given examples to reinforce. To solve them correctly and find different paths from difficult situations, you need to apply the basic property of fractions.

It goes like this: If you multiply both the numerator and the denominator by the same number or expression (other than zero), the value of the common fraction does not change. A special case of this rule is the division of both sides of an expression by the same number or polynomial. Such transformations are called identical equalities.

Below we will look at how to solve addition and subtraction of algebraic fractions, multiplying, dividing and reducing fractions.

Mathematical operations with fractions

Let's look at how to solve, the main property of an algebraic fraction, and how to apply it in practice. If you need to multiply two fractions, add them, divide one by another, or subtract, you must always follow the rules.

Thus, for the operation of addition and subtraction, an additional factor must be found in order to bring the expressions to a common denominator. If the fractions are initially given with the same expressions Q, then this paragraph should be omitted. When common denominator found, how to solve algebraic fractions? You need to add or subtract numerators. But! It must be remembered that if there is a “-” sign in front of the fraction, all signs in the numerator are reversed. Sometimes you should not make any substitutions and mathematical operations. It is enough to change the sign in front of the fraction.

The concept is often used as reducing fractions. This means the following: if the numerator and denominator are divided by an expression different from one (the same for both parts), then a new fraction is obtained. The dividend and divisor are smaller than before, but due to the basic rule of fractions they remain equal to the original example.

The purpose of this operation is to obtain a new irreducible expression. Decide this task You can do this by reducing the numerator and denominator by the greatest common factor. The operation algorithm consists of two points:

Finding gcd for both sides of the fraction.
Dividing the numerator and denominator by the found expression and obtaining an irreducible fraction equal to the previous one.

Below is a table showing the formulas. For convenience, you can print it out and carry it with you in a notebook. However, so that in the future, when solving a test or exam, there will be no difficulties in the question of how to solve algebraic fractions, the indicated formulas must be learned by heart.

Several examples with solutions

WITH theoretical point From a perspective, the question of how to solve algebraic fractions is considered. The examples given in the article will help you better understand the material.

1. Convert fractions and bring them to a common denominator.

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

After studying the theoretical part and considering practical issues there shouldn't be any more.

Division and the numerator and denominator of the fraction on their common divisor, different from one, is called reducing a fraction.

To shorten 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 .

In this article we will look at basic operations with algebraic fractions:

reducing fractions
multiplying fractions
dividing fractions

Let's start with reduction of algebraic fractions.

It would seem that, algorithm obvious.

To reduce algebraic fractions, need to

1. Factor the numerator and denominator of the fraction.

2. Reduce equal factors.

However, schoolchildren often make the mistake of “reducing” not the factors, but the terms. For example, there are amateurs who “reduce” fractions by and get as a result , which, of course, is not true.

Let's look at examples:

1. Reduce fraction:

1. Let us factorize the numerator using the formula of the square of the sum, and the denominator using the formula of the difference of squares

2. Divide the numerator and denominator by

2. Reduce fraction:

1. Let's factorize the numerator. Since the numerator contains four terms, we use grouping.

2. Let's factorize the denominator. We can also use grouping.

3. Let's write down the fraction that we got and reduce the same factors:

Multiplying algebraic fractions.

When multiplying algebraic fractions, we multiply the numerator by the numerator, and multiply the denominator by the denominator.

Important! There is no need to rush to multiply the numerator and denominator of a fraction. After we have written down the product of the numerators of the fractions in the numerator, and the product of the denominators in the denominator, we need to factor each factor and reduce the fraction.

Let's look at examples:

3. Simplify the expression:

1. Let’s write the product of fractions: in the numerator the product of the numerators, and in the denominator the product of the denominators:

2. Let's factorize each bracket:

Now we need to reduce the same factors. Note that the expressions and differ only in sign: and as a result of dividing the first expression by the second we get -1.

So,

We divide algebraic fractions according to the following rule:

That is To divide by a fraction, you need to multiply by the "inverted" one.

We see that dividing fractions comes down to multiplying, and multiplication ultimately comes down to reducing fractions.

Let's look at an example:

4. Simplify the expression: