Reduction of algebraic fractions: a rule, examples. How to solve algebraic fractions? Theory and practice

Fractions and their reduction is another topic that starts in 5th grade. Here the basis of this action is formed, and then these skills are pulled by a thread into higher mathematics. If the student has not learned, then he may have problems in algebra. Therefore, it is better to understand a few rules once and for all. And remember one prohibition and never break it.

Fraction and its reduction

What it is, every student knows. Any two digits located between the horizontal bar are immediately perceived as a fraction. However, not everyone understands that any number can become it. If it is an integer, then it can always be divided by one, then you get an improper fraction. But more on that later.

The beginning is always simple. First you need to figure out how to reduce the correct fraction. That is, one whose numerator is less than the denominator. To do this, you need to remember the main property of a fraction. It states that when multiplying (as well as dividing) its numerator and denominator by the same number at the same time, an equivalent original fraction is obtained.

The division actions that are performed on this property result in a reduction. That is, its maximum simplification. A fraction can be reduced as long as there are common factors above and below the line. When they no longer exist, the reduction is impossible. And they say that this fraction is irreducible.

two ways

1.Step by step reduction. It uses the guessing method, when both numbers are divided by the minimum common factor that the student noticed. If after the first reduction it is clear that this is not the end, then the division continues. Until the fraction becomes irreducible.

2. Finding the greatest common divisor of the numerator and denominator. This is the most rational way to reduce fractions. It involves factoring the numerator and denominator into prime factors. Among them, then you need to choose all the same. Their product will give the largest common factor by which the fraction is reduced.

Both of these methods are equivalent. The student is invited to master them and use the one that he liked best.

What if there are letters and operations of addition and subtraction?

With the first part of the question, everything is more or less clear. Letters can be abbreviated just like numbers. The main thing is that they act as multipliers. But with the second, many have problems.

Important to remember! You can only reduce numbers that are factors. If they are terms, it is impossible.

In order to understand how to reduce fractions that look like an algebraic expression, you need to learn the rule. First, express the numerator and denominator as a product. Then you can reduce if there are common factors. For representation as multipliers, the following tricks are useful:

• grouping;
• bracketing;
• application of abbreviated multiplication identities.

Moreover, the latter method makes it possible to immediately obtain terms in the form of factors. Therefore, it must always be used if a known pattern is visible.

But this is not scary yet, then tasks with degrees and roots appear. That's when you need to muster up the courage and learn a couple of new rules.

Power expression

Fraction. The product in the numerator and denominator. There are letters and numbers. And they are also raised to a power, which also consists of terms or factors. There is something to be afraid of.

In order to figure out how to reduce fractions with powers, you need to learn two points:

• if there is a sum in the exponent, then it can be decomposed into factors, the powers of which will be the original terms;
• if the difference, then into the dividend and the divisor, the first in the degree will be reduced, the second - subtracted.

After completing these steps, the common multipliers become visible. In such examples, it is not necessary to calculate all powers. It is enough to simply reduce the degrees with the same indicators and bases.

In order to finally master how to reduce fractions with powers, you need a lot of practice. After several examples of the same type, the actions will be performed automatically.

What if the expression contains a root?

It can also be shortened. Again, just follow the rules. Moreover, all those described above are true. In general, if the question is how to reduce a fraction with roots, then you need to divide.

It can also be divided into irrational expressions. That is, if the numerator and denominator have the same factors enclosed under the root sign, then they can be safely reduced. This will simplify the expression and get the job done.

If, after the reduction, irrationality remains under the line of the fraction, then you need to get rid of it. In other words, multiply the numerator and denominator by it. If after this operation common factors appeared, then they will need to be reduced again.

That, perhaps, is all about how to reduce fractions. Few rules, but one prohibition. Never reduce terms!

In this article, we will focus on reduction of algebraic fractions. First, let's figure out what is meant by the term "reduction of an algebraic fraction", and find out whether an algebraic fraction is always reducible. Next, we give a rule that allows us to carry out this transformation. Finally, consider the solutions of typical examples that will make it possible to understand all the subtleties of the process.

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What does it mean to reduce an algebraic fraction?

Studying, we talked about their reduction. we called the division of its numerator and denominator by the common factor. For example, the common fraction 30/54 can be reduced by 6 (that is, divided by 6 its numerator and denominator), which will lead us to the fraction 5/9.

The reduction of an algebraic fraction is understood as a similar action. Reduce algebraic fraction is to divide its numerator and denominator by a common factor. But if the common factor of the numerator and denominator of an ordinary fraction can only be a number, then the common factor of the numerator and denominator of an algebraic fraction can be a polynomial, in particular, a monomial or a number.

For example, an algebraic fraction can be reduced by the number 3, which gives the fraction . It is also possible to reduce on the variable x , which will result in the expression . The original algebraic fraction can be reduced by the monomial 3 x, as well as by any of the polynomials x+2 y, 3 x+6 y, x 2 +2 x y or 3 x 2 +6 x y .

The ultimate goal of reducing an algebraic fraction is to obtain a fraction of a simpler form, at best, an irreducible fraction.

Is any algebraic fraction subject to reduction?

We know that ordinary fractions are subdivided into . Irreducible fractions do not have common factors other than unity in the numerator and denominator, therefore, they cannot be reduced.

Algebraic fractions may or may not have common numerator and denominator factors. In the presence of common factors, it is possible to reduce the algebraic fraction. If there are no common factors, then the simplification of the algebraic fraction by means of its reduction is impossible.

In the general case, by the appearance of an algebraic fraction, it is quite difficult to determine whether it is possible to perform its reduction. Undoubtedly, in some cases the common factors of the numerator and denominator are obvious. For example, it is clearly seen that the numerator and denominator of an algebraic fraction have a common factor of 3. It is also easy to see that an algebraic fraction can be reduced by x, by y, or immediately by x·y. But much more often, the common factor of the numerator and denominator of an algebraic fraction is not immediately visible, and even more often, it simply does not exist. For example, a fraction can be reduced by x−1 , but this common factor is clearly not present in the notation. And an algebraic fraction cannot be reduced because its numerator and denominator do not have common factors.

In general, the question of the contractibility of an algebraic fraction is very difficult. And sometimes it’s easier to solve a problem by working with an algebraic fraction in its original form than to find out if this fraction can be preliminarily reduced. But still, there are transformations that in some cases allow, with relatively little effort, to find the common factors of the numerator and denominator, if any, or to conclude that the original algebraic fraction is irreducible. This information will be disclosed in the next paragraph.

Algebraic fraction reduction rule

The information of the previous paragraphs allows you to naturally perceive the following algebraic fraction reduction rule, which consists of two steps:

• first, the common factors of the numerator and denominator of the original fraction are found;
• if any, then reduction by these factors is carried out.

These steps of the announced rule need clarification.

The most convenient way to find common ones is to factorize the polynomials that are in the numerator and denominator of the original algebraic fraction. In this case, the common factors of the numerator and denominator immediately become visible, or it becomes clear that there are no common factors.

If there are no common factors, then we can conclude that the algebraic fraction is irreducible. If the common factors are found, then at the second step they are reduced. The result is a new fraction of a simpler form.

The rule of reduction of algebraic fractions is based on the main property of an algebraic fraction, which is expressed by the equality , where a , b and c are some polynomials, and b and c are non-zero. At the first step, the original algebraic fraction is reduced to the form , from which the common factor c becomes visible, and at the second step, reduction is performed - the transition to the fraction .

Let's move on to solving examples using this rule. On them, we will analyze all the possible nuances that arise when decomposing the numerator and denominator of an algebraic fraction into factors and subsequent reduction.

Typical examples

First you need to say about the reduction of algebraic fractions, the numerator and denominator of which are the same. Such fractions are identically equal to one on the entire ODZ of the variables included in it, for example,
etc.

Now it does not hurt to remember how the reduction of ordinary fractions is performed - after all, they are a special case of algebraic fractions. Natural numbers in the numerator and denominator of an ordinary fraction, after which the common factors are reduced (if any). For example, . The product of identical prime factors can be written in the form of degrees, and when reduced, use. In this case, the solution would look like this: , here we divided the numerator and denominator by a common factor 2 2 3 . Or, for greater clarity, based on the properties of multiplication and division, the solution is presented in the form.

According to absolutely similar principles, the reduction of algebraic fractions is carried out, in the numerator and denominator of which there are monomials with integer coefficients.

Example.

Reduce algebraic fraction .

Solution.

You can represent the numerator and denominator of the original algebraic fraction as a product of simple factors and variables, and then carry out the reduction:

But it is more rational to write the solution as an expression with powers:

Answer:

.

As for the reduction of algebraic fractions that have fractional numerical coefficients in the numerator and denominator, you can do two things: either separately divide these fractional coefficients, or first get rid of fractional coefficients by multiplying the numerator and denominator by some natural number. We talked about the last transformation in the article bringing an algebraic fraction to a new denominator, it can be carried out due to the main property of an algebraic fraction. Let's deal with this with an example.

Example.

Perform fraction reduction.

Solution.

You can reduce the fraction like this: .

And it was possible to get rid of fractional coefficients first by multiplying the numerator and denominator by the denominators of these coefficients, that is, by LCM(5, 10)=10 . In this case we have .

Answer:

.

You can move on to algebraic fractions of a general form, in which the numerator and denominator can contain both numbers and monomials, as well as polynomials.

When reducing such fractions, the main problem is that the common factor of the numerator and denominator is not always visible. Moreover, it does not always exist. In order to find a common factor or make sure that it does not exist, you need to factorize the numerator and denominator of an algebraic fraction.

Example.

Reduce rational fraction .

Solution.

To do this, we factorize the polynomials in the numerator and denominator. Let's start with parentheses: . Obviously, parenthesized expressions can be converted using

Based on their main property: if the numerator and denominator of a fraction are divided by the same non-zero polynomial, then a fraction equal to it will be obtained.

You can only reduce multipliers!

Members of polynomials cannot be reduced!

To reduce an algebraic fraction, the polynomials in the numerator and denominator must first be factored.

Consider examples of fraction reduction.

The numerator and denominator of a fraction are monomials. They represent work(numbers, variables and their degrees), multipliers we can reduce.

We reduce the numbers by their greatest common divisor, that is, by the largest number by which each of the given numbers is divisible. For 24 and 36, this is 12. After the reduction from 24, 2 remains, from 36 - 3.

We reduce the degrees by the degree with the smallest indicator. To reduce a fraction means to divide the numerator and denominator by the same divisor, and subtract the exponents.

a² and a⁷ are reduced by a². At the same time, one remains in the numerator from a² (we write 1 only if there are no other factors left after reduction. 2 remains from 24, so we do not write the 1 remaining from a²). From a⁷ after reduction remains a⁵.

b and b are abbreviated by b, the resulting units are not written.

c³º and c⁵ are reduced by c⁵. From c³º, c²⁵ remains, from c⁵ - unit (we do not write it). In this way,

The numerator and denominator of this algebraic fraction are polynomials. It is impossible to reduce the terms of polynomials! (cannot be reduced, for example, 8x² and 2x!). To reduce this fraction, it is necessary. The numerator has a common factor of 4x. Let's take it out of brackets:

Both the numerator and the denominator have the same factor (2x-3). We reduce the fraction by this factor. We got 4x in the numerator, 1 in the denominator. According to 1 property of algebraic fractions, the fraction is 4x.

You can only reduce factors (you cannot reduce a given fraction by 25x²!). Therefore, the polynomials in the numerator and denominator of a fraction must be factored.

The numerator is the full square of the sum, and the denominator is the difference of the squares. After expansion by the formulas of abbreviated multiplication, we get:

We reduce the fraction by (5x + 1) (to do this, cross out the two in the numerator as an exponent, from (5x + 1) ² this will leave (5x + 1)):

The numerator has a common factor of 2, let's take it out of brackets. In the denominator - the formula for the difference of cubes:

As a result of expansion in the numerator and denominator, we got the same factor (9 + 3a + a²). We reduce the fraction on it:

The polynomial in the numerator consists of 4 terms. the first term with the second, the third with the fourth, and we take out the common factor x² from the first brackets. We decompose the denominator according to the formula for the sum of cubes:

In the numerator, we take out the common factor (x + 2) out of brackets:

We reduce the fraction by (x + 2):

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

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Meaning of Algebraic Fraction Abbreviation

In the materials on the ordinary fraction, we considered its reduction. We have defined the reduction of a common fraction as dividing its numerator and denominator by a common factor.

Reducing an algebraic fraction is a similar operation.

Definition 1

Algebraic fraction reduction is the division of its numerator and denominator by a common factor. In this case, unlike the reduction of an ordinary fraction (only a number can be a common denominator), a polynomial, in particular, a monomial or a number, can serve as a common factor for the numerator and denominator of an algebraic fraction.

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, as a result we get: 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 the materials on ordinary fractions, we know that there are reducible and irreducible fractions. Irreducible - these are fractions that do not have common factors of the numerator and denominator, other than 1.

With algebraic fractions, everything is the same: they may or may not have common factors of the numerator and denominator. 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 by the reduction method.

In general cases, for a given type of fraction, it is quite difficult to understand whether it is subject to reduction. Of course, in some cases, the presence of a common factor of 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 a fraction - x · y 5 · x · y · z 3 we also immediately understand that it is possible to reduce it by x, or y, or by x · y. And yet, examples of algebraic fractions are much more common, 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 in the record. 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 finding out the contractibility 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 contractible. In this case, there are such transformations that in particular cases allow us to determine the common factor of the numerator and denominator or to conclude that the fraction is irreducible. We will analyze this issue in detail in the next paragraph of the article.

Algebraic fraction reduction rule

Algebraic fraction reduction rule consists of two consecutive steps:

• finding the common factors of the numerator and denominator;
• in the case of finding such, the implementation of the direct action of reducing the fraction.

The most convenient method for finding common denominators is to factorize the polynomials present in the numerator and denominator of a given algebraic fraction. This allows you to immediately visually 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 the reduction, i.e. transition to a fraction of the form a b .

Typical examples

Despite some obviousness, let's clarify about 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 decomposed into prime factors, then the common factors are reduced (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 degrees, and in the process of fraction reduction, use the property of dividing degrees with the same 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 will 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

Given an algebraic fraction - 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 prime factors and variables, and then reduce:

27 a 5 b 2 c z 6 a 2 b 2 c 7 z = - 3 3 3 a a a a a a b b c z 2 3 a a b b c c c c c c c c z = = - 3 3 a a a 2 c 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 cc 7 zz = = - 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 there are fractional numerical coefficients in the numerator and denominator of an algebraic fraction, there are two possible ways of further action: either separately divide these fractional coefficients, 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 main property of an algebraic fraction (you can read about it in the article “Reducing an algebraic fraction to a new denominator”).

Example 2

Given a fraction 2 5 x 0 , 3 x 3 . It needs to be reduced.

Solution

It is possible to reduce the fraction in 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 previously got rid of fractional coefficients - we multiply the numerator and denominator by the least common multiple of the denominators of these coefficients, i.e. per 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 both monomials and polynomials, a problem is possible when the common factor is not always immediately visible. Or more than that, it simply doesn't exist. Then, to determine the common factor or fix the fact of its absence, the numerator and denominator of the algebraic fraction are factorized.

Example 3

Given a rational fraction 2 · a 2 · b 2 + 28 · a · b 2 + 98 · b 2 a 2 · b 3 - 49 · b 3 . It needs to be shortened.

Solution

Let us factorize the polynomials in the numerator and denominator. Let's do the parentheses:

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 brackets can be converted using the 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 the 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

We 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 the common factors are hidden by numerical coefficients. Then, when reducing fractions, it is optimal to take out the numerical factors at higher powers of the numerator and denominator.

Example 4

Given an algebraic fraction 1 5 x - 2 7 x 3 y 5 x 2 y - 3 1 2 . It should be reduced 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 at 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 multiplier 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 factorize polynomials.

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At first glance, algebraic fractions seem very complicated, and an unprepared student may think that it is impossible to do anything with them. The piling up of variables, numbers, and even powers inspires fear. However, the same rules are used to reduce fractions (such as 15/25) and algebraic fractions.

Steps

Fraction reduction

Learn how to work with simple fractions. Operations with ordinary and algebraic fractions are similar. For example, take the fraction 15/35. To simplify this fraction, find a common divisor. Both numbers are divisible by five, so we can extract 5 in the numerator and denominator:

15 → 5 * 3 35 → 5 * 7

Now you can reduce common factors, that is, cross out the 5 in the numerator and denominator. As a result, we get a simplified fraction 3/7 . In algebraic expressions, common factors are distinguished in the same way as in ordinary ones. In the previous example, we were able to easily extract 5 out of 15 - the same principle applies to more complex expressions such as 15x - 5. Let's find the common factor. In this case, it will be 5, since both terms (15x and -5) are divisible by 5. As before, we select the common factor and transfer it to the left.

15x - 5 = 5 * (3x - 1)

To check if everything is correct, it is enough to multiply the expression in brackets by 5 - the result will be the same numbers that were at first. Complex terms can be distinguished in the same way as simple ones. For algebraic fractions, the same principles apply as for ordinary fractions. This is the easiest way to reduce a fraction. Consider the following fraction:

(x+2)(x-3)(x+2)(x+10)

Note that both the numerator (top) and denominator (bottom) have a term (x+2), so it can be reduced in the same way as the common factor 5 in 15/35:

(x+2) (x-3) → (x-3)(x+2) (x+10) → (x+10)

As a result, we get a simplified expression: (x-3)/(x+10)

Reduction of algebraic fractions

Find the common factor in the numerator, that is, at the top of the fraction. When reducing an algebraic fraction, the first step is to simplify both of its parts. Start with the numerator and try to factor it into as many factors as possible. Consider in this section the following fraction:

9x-3 15x+6

Let's start with the numerator: 9x - 3. For 9x and -3, the common factor is the number 3. Let's take 3 out of brackets, as we do with ordinary numbers: 3 * (3x-1). As a result of this transformation, the following fraction will be obtained:

3(3x-1) 15x+6

Find the common factor in the numerator. Let's continue the execution of the above example and write out the denominator: 15x+6. As before, we find by what number both parts are divisible. And in this case the common factor is 3, so we can write: 3 * (5x +2). Let's rewrite the fraction in the following form:

3(3x-1) 3(5x+2)

Reduce identical terms. In this step, you can simplify the fraction. Cancel the same terms in the numerator and denominator. In our example, this number is 3.

3 (3x-1) → (3x-1) 3 (5x+2) → (5x+2)

Determine that the fraction has the simplest form. A fraction is completely simplified when there are no common factors left in the numerator and denominator. Note that you cannot abbreviate those terms that are inside the brackets - in the above example, there is no way to extract x from 3x and 5x, since (3x -1) and (5x + 2) are full members. Thus, the fraction is not amenable to further simplification, and the final answer is as follows:

(3x-1)(5x+2)

Practice reducing fractions yourself. The best way to learn the method is to solve problems on your own. The correct answers are given below the examples.

4(x+2)(x-13)(4x+8)

Answer:(x=13)

2x 2-x 5x

Answer:(2x-1)/5

Special Moves

Move the negative sign out of the fraction. Suppose we are given the following fraction:

3(x-4) 5(4x)

Note that (x-4) and (4-x) are “nearly” identical, but they cannot be canceled outright because they are “flipped”. However, (x - 4) can be written as -1 * (4 - x), just as (4 + 2x) can be written as 2 * (2 + x). This is called "sign reversal".

-1*3(4-x) 5(4x)

Now you can reduce the same terms (4-x):

-1 * 3 (4-x) 5 (4x)

So here is the final answer: -3/5 . Learn to recognize the difference of squares. The difference of squares is when the square of one number is subtracted from the square of another number, as in the expression (a 2 - b 2). The difference of perfect squares can always be decomposed into two parts - the sum and the difference of the corresponding square roots. Then the expression will take the following form:

A 2 - b 2 = (a+b)(a-b)

This trick is very useful when looking for common terms in algebraic fractions.

• Check if you have correctly factored this or that expression. To do this, multiply the factors - the result should be the same expression.
• To completely simplify a fraction, always select the largest factors.