Rules for reducing fractions with unknowns. Reducing algebraic fractions

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. We reason 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 . Next, 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 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 a 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 a 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:

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), then the value common fraction won't change. A special case from 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 you need to omit this item. Once the common denominator is found, how do you 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 perform any substitutions or 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. You can solve this problem by reducing the numerator and denominator by the greatest common factor. The operation algorithm consists of two points:

1. Finding gcd for both sides of the fraction.
2. 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, these formulas must be learned by heart.

Several examples with solutions

From a theoretical point of view, 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.

It is based on their main 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. 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 . 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):

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. From the names you can guess that reducible fractions can be reduced, but irreducible fractions cannot.

What does it mean to reduce a fraction? Reduce a 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 greatest common divisor of the numerator and denominator of a fraction is 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's a summary of the solution: .

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