Transformation of rational expressions: types of transformations, examples. Transformation of rational (algebraic) fractions, types of transformations, examples Transformation of fractionally rational algebraic expressions

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Converting Rational Expressions

This paragraph sums up everything we've said since 7th grade about mathematical language, mathematical symbolism, numbers, variables, powers, polynomials, and algebraic fractions. But first, let's take a short digression into the past.

Remember how things were with the study of numbers and numerical expressions in the lower grades.

And, say, only one label can be attached to a fraction - a rational number.

The situation is similar with algebraic expressions: the first stage of their study is numbers, variables, degrees (“numbers”); the second stage of their study is monomials (“natural numbers”); the third stage of their study is polynomials ("whole numbers"); the fourth stage of their study - algebraic fractions
("rational numbers"). Moreover, each next stage, as it were, absorbs the previous one: for example, numbers, variables, degrees are special cases of monomials; monomials are special cases of polynomials; polynomials are special cases of algebraic fractions. By the way, the following terms are sometimes used in algebra: a polynomial is an integer expression, an algebraic fraction is a fractional expression (this only strengthens the analogy).

Let's continue with the above analogy. You know that any numeric expression, after performing all the arithmetic operations included in it, takes on a specific numerical value - a rational number (of course, it can turn out to be a natural number, an integer, or a fraction - it doesn't matter). Similarly, any algebraic expression composed of numbers and variables using arithmetic operations and raising to a natural degree, after transformations, it takes the form of an algebraic fraction and again, in particular, it may turn out not to be a fraction, but a polynomial or even a monomial). For such expressions in algebra, the term rational expression is used.

Example. Prove Identity

Decision.
To prove an identity means to establish that for all admissible values of the variables, its left and right parts are identically equal expressions. In algebra, identities are proved in various ways:

1) perform transformations of the left side and get the right side as a result;

2) perform transformations of the right side and get the left side as a result;

3) separately convert the right and left parts and get the same expression in the first and second cases;

4) make up the difference between the left and right parts and, as a result of its transformations, get zero.

Which method to choose depends on the specific type identities which you are asked to prove. In this example, it is advisable to choose the first method.

To convert rational expressions, the same procedure is adopted as for converting numeric expressions. This means that first the actions in brackets are performed, then the actions of the second stage (multiplication, division, exponentiation), then the actions of the first stage (addition, subtraction).

Let's perform transformations by actions, based on those rules, algorithms that have been developed in the previous paragraphs.

As you can see, we managed to transform the left side of the identity under test to the form of the right side. This means that the identity has been proven. However, we recall that the identity is valid only for admissible values of the variables. Those in this example are any values of a and b, except for those that turn the denominators of fractions to zero. This means that any pairs of numbers (a; b) are admissible, except for those for which at least one of the equalities is satisfied:

2a - b = 0, 2a + b = 0, b = 0.

Mordkovich A. G., Algebra. Grade 8: Proc. for general education institutions. - 3rd ed., finalized. - M.: Mnemosyne, 2001. - 223 p.: ill.

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From the algebra course of the school curriculum, we turn to the specifics. In this article, we will study in detail a special kind of rational expressions − rational fractions, and also analyze what characteristic identical transformations of rational fractions take place.

We note right away that rational fractions in the sense in which we define them below are called algebraic fractions in some algebra textbooks. That is, in this article we will understand the same thing under rational and algebraic fractions.

As usual, we start with a definition and examples. Next, let's talk about bringing a rational fraction to a new denominator and about changing the signs of the members of the fraction. After that, we will analyze how the reduction of fractions is performed. Finally, let us dwell on the representation of a rational fraction as a sum of several fractions. All information will be provided with examples with detailed descriptions of solutions.

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Definition and examples of rational fractions

Rational fractions are studied in algebra lessons in grade 8. We will use the definition of a rational fraction, which is given in the algebra textbook for grades 8 by Yu. N. Makarychev and others.

This definition does not specify whether the polynomials in the numerator and denominator of a rational fraction must be polynomials of standard form or not. Therefore, we will assume that rational fractions can contain both standard and non-standard polynomials.

Here are a few examples of rational fractions. So , x/8 and - rational fractions. And fractions and do not fit the sounded definition of a rational fraction, since in the first of them the numerator is not a polynomial, and in the second both the numerator and the denominator contain expressions that are not polynomials.

Converting the numerator and denominator of a rational fraction

The numerator and denominator of any fraction are self-sufficient mathematical expressions, in the case of rational fractions they are polynomials, in a particular case they are monomials and numbers. Therefore, with the numerator and denominator of a rational fraction, as with any expression, identical transformations can be carried out. In other words, the expression in the numerator of a rational fraction can be replaced by an expression that is identically equal to it, just like the denominator.

In the numerator and denominator of a rational fraction, identical transformations can be performed. For example, in the numerator, you can group and reduce similar terms, and in the denominator, the product of several numbers can be replaced by its value. And since the numerator and denominator of a rational fraction are polynomials, it is possible to perform transformations characteristic of polynomials with them, for example, reduction to a standard form or representation as a product.

For clarity, consider the solutions of several examples.

Example.

Convert Rational Fraction so that the numerator is a polynomial of the standard form, and the denominator is the product of polynomials.

Decision.

Reducing rational fractions to a new denominator is mainly used when adding and subtracting rational fractions.

Changing signs in front of a fraction, as well as in its numerator and denominator

The basic property of a fraction can be used to change the signs of the terms of the fraction. Indeed, multiplying the numerator and denominator of a rational fraction by -1 is tantamount to changing their signs, and the result is a fraction that is identically equal to the given one. Such a transformation has to be used quite often when working with rational fractions.

Thus, if you simultaneously change the signs of the numerator and denominator of a fraction, you will get a fraction equal to the original one. This statement corresponds to equality.

Let's take an example. A rational fraction can be replaced by an identically equal fraction with reversed signs of the numerator and denominator of the form.

With fractions, one more identical transformation can be carried out, in which the sign is changed either in the numerator or in the denominator. Let's go over the appropriate rule. If you replace the sign of a fraction together with the sign of the numerator or denominator, you get a fraction that is identically equal to the original one. The written statement corresponds to the equalities and .

It is not difficult to prove these equalities. The proof is based on the properties of multiplication of numbers. Let's prove the first of them: . With the help of similar transformations, the equality is also proved.

For example, a fraction can be replaced by an expression or .

To conclude this subsection, we present two more useful equalities and . That is, if you change the sign of only the numerator or only the denominator, then the fraction will change its sign. For example, and .

The considered transformations, which allow changing the sign of the terms of a fraction, are often used when transforming fractionally rational expressions.

Reduction of rational fractions

The following transformation of rational fractions, called the reduction of rational fractions, is based on the same basic property of a fraction. This transformation corresponds to the equality , where a , b and c are some polynomials, and b and c are non-zero.

From the above equality, it becomes clear that the reduction of a rational fraction implies getting rid of the common factor in its numerator and denominator.

Example.

Reduce the rational fraction.

Decision.

The common factor 2 is immediately visible, let's reduce it (when writing, it is convenient to cross out the common factors by which the reduction is made). We have . Since x 2 \u003d x x and y 7 \u003d y 3 y 4 (see if necessary), it is clear that x is a common factor of the numerator and denominator of the resulting fraction, like y 3 . Let's reduce by these factors: . This completes the reduction.

Above, we performed the reduction of a rational fraction sequentially. And it was possible to perform the reduction in one step, immediately reducing the fraction by 2·x·y 3 . In this case, the solution would look like this: .

Answer:

When reducing rational 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 a rational fraction. If there is no common factor, then the original rational fraction does not need to be reduced, otherwise, the reduction is carried out.

In the process of reducing rational fractions, various nuances may arise. The main subtleties with examples and details are discussed in the article reduction of algebraic fractions.

Concluding the conversation about the reduction of rational fractions, we note that this transformation is identical, and the main difficulty in its implementation lies in the factorization of polynomials in the numerator and denominator.

Representation of a rational fraction as a sum of fractions

Quite specific, but in some cases very useful, is the transformation of a rational fraction, which consists in its representation as the sum of several fractions, or the sum of an integer expression and a fraction.

A rational fraction, in the numerator of which there is a polynomial, which is the sum of several monomials, can always be written as the sum of fractions with the same denominators, in the numerators of which are the corresponding monomials. For example, . This representation is explained by the rule of addition and subtraction of algebraic fractions with the same denominators.

In general, any rational fraction can be represented as a sum of fractions in many different ways. For example, the fraction a/b can be represented as the sum of two fractions - an arbitrary fraction c/d and a fraction equal to the difference between the fractions a/b and c/d. This statement is true, since the equality . For example, a rational fraction can be represented as a sum of fractions in various ways: We represent the original fraction as the sum of an integer expression and a fraction. After dividing the numerator by the denominator by a column, we get the equality . The value of the expression n 3 +4 for any integer n is an integer. And the value of a fraction is an integer if and only if its denominator is 1, −1, 3, or −3. These values correspond to the values n=3 , n=1 , n=5 and n=−1 respectively.

Answer:

−1 , 1 , 3 , 5 .

Bibliography.

Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
Mordkovich A. G. Algebra. 7th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 13th ed., Rev. - M.: Mnemosyne, 2009. - 160 p.: ill. ISBN 978-5-346-01198-9.
Mordkovich A. G. Algebra. 8th grade. At 2 pm Part 1. A textbook for students of educational institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemozina, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Rational expressions and fractions are the cornerstone of the entire course of algebra. Those who learn how to work with such expressions, simplify them and factor them, in fact, will be able to solve any problem, since the transformation of expressions is an integral part of any serious equation, inequality, and even a word problem.

In this video tutorial, we'll see how to correctly apply abbreviated multiplication formulas to simplify rational expressions and fractions. Let's learn to see these formulas where, at first glance, there is nothing. At the same time, we repeat such a simple trick as factoring a square trinomial into factors through the discriminant.

As you probably already guessed from the formulas behind my back, today we will study the formulas for abbreviated multiplication, or rather, not the formulas themselves, but their application to simplify and reduce complex rational expressions. But, before moving on to solving examples, let's take a closer look at these formulas or recall them:

$((a)^(2))-((b)^(2))=\left(a-b \right)\left(a+b \right)$ is the difference of squares;
$((\left(a+b \right))^(2))=((a)^(2))+2ab+((b)^(2))$ is the square of the sum;
$((\left(a-b \right))^(2))=((a)^(2))-2ab+((b)^(2))$ is the squared difference;
$((a)^(3))+((b)^(3))=\left(a+b \right)\left(((a)^(2))-ab+((b)^( 2)) \right)$ is the sum of cubes;
$((a)^(3))-((b)^(3))=\left(a-b \right)\left(((a)^(2))+ab+((b)^(2) ) \right)$ is the difference of cubes.

I would also like to note that our school education system is designed in such a way that it is with the study of this topic, i.e. rational expressions, as well as roots, modules, all students have the same problem, which I will now explain.

The fact is that at the very beginning of studying the formulas for abbreviated multiplication and, accordingly, actions to reduce fractions (this is about grade 8), teachers say something like this: “If something is not clear to you, then do not worry, we will we will return to this topic more than once, in high school for sure. We'll figure it out later." Well, then, at the turn of grades 9-10, the same teachers explain to the same students who still don’t know how to solve rational fractions, something like this: “Where were you the previous two years? The same was studied in algebra in the 8th grade! What can be incomprehensible here? It's so obvious!"

However, for ordinary students, such explanations are not at all easier: they still had a mess in their heads, so right now we will analyze two simple examples, on the basis of which we will see how to highlight these expressions in real problems, which will lead us to short multiplication formulas and how to apply it later to transform complex rational expressions.

Reduction of simple rational fractions

Task #1

\[\frac(4x+3((y)^(2)))(9((y)^(4))-16((x)^(2)))\]

The first thing we need to learn is to distinguish exact squares and higher powers in the original expressions, on the basis of which we can then apply the formulas. Let's get a look:

Let's rewrite our expression taking into account these facts:

\[\frac(4x+3((y)^(2)))(((\left(3((y)^(2)) \right))^(2))-((\left(4x \right))^(2)))=\frac(4x+3((y)^(2)))(\left(3((y)^(2))-4x \right)\left(3 ((y)^(2))+4x \right))=\frac(1)(3((y)^(2))-4x)\]

Answer: $\frac(1)(3((y)^(2))-4x)$.

Task #2

Let's move on to the second task:

\[\frac(8)(((x)^(2))+5xy-6((y)^(2)))\]

There is nothing to simplify here, because the numerator is a constant, but I proposed this problem precisely so that you learn how to factorize polynomials containing two variables. If instead of it there was a polynomial written below, how would we decompose it?

\[((x)^(2))+5x-6=\left(x-... \right)\left(x-... \right)\]

Let's solve the equation and find $x$ that we can put in place of dots:

\[((x)^(2))+5x-6=0\]

\[((x)_(1))=\frac(-5+7)(2)=\frac(2)(2)=1\]

\[((x)_(2))=\frac(-5-7)(2)=\frac(-12)(2)=-6\]

We can rewrite the trinomial as follows:

\[((x)^(2))+5xy-6((y)^(2))=\left(x-1 \right)\left(x+6 \right)\]

We learned how to work with a square trinomial - for this we had to record this video lesson. But what if, in addition to $x$ and the constant, there is also $y$? Let's look at them as another element of the coefficients, i.e. Let's rewrite our expression as follows:

\[((x)^(2))+5y\cdot x-6((y)^(2))\]

\[((x)_(1))=\frac(-5y+7y)(2)=y\]

\[((x)_(2))=\frac(-5y-7y)(2)=\frac(-12y)(2)=-6y\]

We write the decomposition of our square construction:

\[\left(x-y \right)\left(x+6y \right)\]

In total, if we return to the original expression and rewrite it taking into account the changes, we get the following:

\[\frac(8)(\left(x-y \right)\left(x+6y \right))\]

What does such a record give us? Nothing, because it cannot be reduced, it is not multiplied or divided by anything. However, as soon as this fraction turns out to be an integral part of a more complex expression, such an expansion will come in handy. Therefore, as soon as you see a square trinomial (whether it is burdened with additional parameters or not), always try to factor it.

Nuances of the solution

Remember the basic rules for converting rational expressions:

All denominators and numerators must be factored either through abbreviated multiplication formulas or through the discriminant.
We need to work according to this algorithm: when we look and try to highlight the abbreviated multiplication formula, then, first of all, we try to translate everything to the maximum possible degree. After that, we take the general degree out of brackets.
Very often there will be expressions with a parameter: other variables will appear as coefficients. We find them using the quadratic expansion formula.

Thus, as soon as you see rational fractions, the first thing to do is to factor both the numerator and denominator into factors (into linear expressions), while we use the reduced multiplication formulas or the discriminant.

Let's look at a couple of such rational expressions and try to factor them out.

Solving More Complex Examples

Task #1

\[\frac(4((x)^(2))-6xy+9((y)^(2)))(2x-3y)\cdot \frac(9((y)^(2))- 4((x)^(2)))(8((x)^(3))+27((y)^(3)))\]

We rewrite and try to expand each term:

Let's rewrite our entire rational expression with these facts in mind:

\[\frac(((\left(2x \right))^(2))-2x\cdot 3y+((\left(3y \right))^(2)))(2x-3y)\cdot \frac (((\left(3y \right))^(2))-((\left(2x \right))^(2)))(((\left(2x \right))^(3))+ ((\left(3y\right))^(3)))=\]

\[=\frac(((\left(2x \right))^(2))-2x\cdot 3y+((\left(3y \right))^(2)))(2x-3y)\cdot \ frac(\left(3y-2x \right)\left(3y+2x \right))(\left(2x+3y \right)\left(((\left(2x \right))^(2))- 2x\cdot 3y+((\left(3y \right))^(2)) \right))=-1\]

Answer: $-1$.

Task #2

\[\frac(3-6x)(2((x)^(2))+4x+8)\cdot \frac(2x+1)(((x)^(2))+4-4x)\ cdot \frac(8-((x)^(3)))(4((x)^(2))-1)\]

Let's look at all fractions.

\[((x)^(2))+4-4x=((x)^(2))-4x+2=((x)^(2))-2\cdot 2x+((2)^( 2))=((\left(x-2 \right))^(2))\]

Let's rewrite the whole structure taking into account the changes:

\[\frac(3\left(1-2x \right))(2\left(((x)^(2))+2x+((2)^(2)) \right))\cdot \frac( 2x+1)(((\left(x-2 \right))^(2)))\cdot \frac(\left(2-x \right)\left(((2)^(2))+ 2x+((x)^(2)) \right))(\left(2x-1 \right)\left(2x+1 \right))=\]

\[=\frac(3\cdot \left(-1 \right))(2\cdot \left(x-2 \right)\cdot \left(-1 \right))=\frac(3)(2 \left(x-2 \right))\]

Answer: $\frac(3)(2\left(x-2 \right))$.

Nuances of the solution

So what have we just learned:

Not every square trinomial is factorized, in particular, this applies to the incomplete square of the sum or difference, which are very often found as parts of the sum or difference cubes.
Constants, i.e. ordinary numbers that do not have variables with them can also act as active elements in the decomposition process. Firstly, they can be taken out of brackets, and secondly, the constants themselves can be represented as powers.
Very often, after decomposing all elements into factors, opposite constructions arise. You need to reduce these fractions very carefully, because when you cross them out either from above or from below, an additional factor $-1$ appears - this is precisely the consequence of the fact that they are opposite.

Solving complex problems

\[\frac(27((a)^(3))-64((b)^(3)))(((b)^(2))-4):\frac(9((a)^ (2))+12ab+16((b)^(2)))(((b)^(2))+4b+4)\]

Let's consider each term separately.

First fraction:

\[((\left(3a \right))^(3))-((\left(4b \right))^(3))=\left(3a-4b \right)\left(((\left (3a \right))^(2))+3a\cdot 4b+((\left(4b \right))^(2)) \right)\]

\[((b)^(2))-((2)^(2))=\left(b-2 \right)\left(b+2 \right)\]

We can rewrite the entire numerator of the second fraction as follows:

\[((\left(3a \right))^(2))+3a\cdot 4b+((\left(4b \right))^(2))\]

Now let's look at the denominator:

\[((b)^(2))+4b+4=((b)^(2))+2\cdot 2b+((2)^(2))=((\left(b+2 \right ))^(2))\]

Let's rewrite the entire rational expression with the above facts in mind:

\[\frac(\left(3a-4b \right)\left(((\left(3a \right))^(2))+3a\cdot 4b+((\left(4b \right))^(2 )) \right))(\left(b-2 \right)\left(b+2 \right))\cdot \frac(((\left(b+2 \right))^(2)))( ((\left(3a \right))^(2))+3a\cdot 4b+((\left(4b \right))^(2)))=\]

\[=\frac(\left(3a-4b \right)\left(b+2 \right))(\left(b-2 \right))\]

Answer: $\frac(\left(3a-4b \right)\left(b+2 \right))(\left(b-2 \right))$.

Nuances of the solution

As we have seen once again, incomplete squares of the sum or incomplete squares of the difference, which are often found in real rational expressions, however, do not be afraid of them, because after the transformation of each element they almost always cancel. In addition, in no case should you be afraid of large constructions in the final answer - it is quite possible that this is not your mistake (especially if everything is factored), but the author conceived such an answer.

In conclusion, I would like to analyze one more complex example, which is no longer directly related to rational fractions, but it contains everything that awaits you on real tests and exams, namely: factorization, reduction to a common denominator, reduction of similar terms. That's exactly what we're going to do now.

Solving a complex problem of simplifying and transforming rational expressions

\[\left(\frac(x)(((x)^(2))+2x+4)+\frac(((x)^(2))+8)(((x)^(3) )-8)-\frac(1)(x-2) \right)\cdot \left(\frac(((x)^(2)))(((x)^(2))-4)- \frac(2)(2-x) \right)\]

First, consider and expand the first bracket: in it we see three separate fractions with different denominators, so the first thing we need to do is bring all three fractions to a common denominator, and for this, each of them should be factored:

\[((x)^(2))+2x+4=((x)^(2))+2\cdot x+((2)^(2))\]

\[((x)^(2))-8=((x)^(3))-((2)^(2))=\left(x-2 \right)\left(((x) ^(2))+2x+((2)^(2)) \right)\]

Let's rewrite our entire structure as follows:

\[\frac(x)(((x)^(2))+2x+((2)^(2)))+\frac(((x)^(2))+8)(\left(x -2 \right)\left(((x)^(2))+2x+((2)^(2)) \right))-\frac(1)(x-2)=\]

\[=\frac(x\left(x-2 \right)+((x)^(3))+8-\left(((x)^(2))+2x+((2)^(2 )) \right))(\left(x-2 \right)\left(((x)^(2))+2x+((2)^(2)) \right))=\]

\[=\frac(((x)^(2))-2x+((x)^(2))+8-((x)^(2))-2x-4)(\left(x-2 \right)\left(((x)^(2))+2x+((2)^(2)) \right))=\frac(((x)^(2))-4x-4)(\ left(x-2 \right)\left(((x)^(2))+2x+((2)^(2)) \right))=\]

\[=\frac(((\left(x-2 \right))^(2)))(\left(x-2 \right)\left(((x)^(2))+2x+(( 2)^(2)) \right))=\frac(x-2)(((x)^(2))+2x+4)\]

This is the result of the calculations from the first parenthesis.

Dealing with the second parenthesis:

\[((x)^(2))-4=((x)^(2))-((2)^(2))=\left(x-2 \right)\left(x+2 \ right)\]

Let's rewrite the second bracket, taking into account the changes:

\[\frac(((x)^(2)))(\left(x-2 \right)\left(x+2 \right))+\frac(2)(x-2)=\frac( ((x)^(2))+2\left(x+2 \right))(\left(x-2 \right)\left(x+2 \right))=\frac(((x)^ (2))+2x+4)(\left(x-2 \right)\left(x+2 \right))\]

Now let's write the entire original construction:

\[\frac(x-2)(((x)^(2))+2x+4)\cdot \frac(((x)^(2))+2x+4)(\left(x-2 \right)\left(x+2 \right))=\frac(1)(x+2)\]

Answer: $\frac(1)(x+2)$.

Nuances of the solution

As you can see, the answer turned out to be quite sane. However, please note: very often with such large-scale calculations, when the only variable is only in the denominator, students forget that this is the denominator and it should be at the bottom of the fraction and write this expression in the numerator - this is a gross mistake.

In addition, I would like to draw your special attention to how such tasks are formalized. In any complex calculations, all steps are performed step by step: first, we count the first bracket separately, then the second bracket separately, and only at the end do we combine all the parts and calculate the result. Thus, we insure ourselves against stupid mistakes, carefully write down all the calculations and at the same time do not waste any extra time, as it might seem at first glance.

This article is about transformation of rational expressions, mostly fractionally rational, is one of the key questions of the algebra course for grades 8. First, we recall what kind of expressions are called rational. Next, we will focus on performing standard transformations with rational expressions, such as grouping terms, taking common factors out of brackets, reducing similar terms, etc. Finally, we will learn how to represent fractional rational expressions as rational fractions.

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Definition and examples of rational expressions

Rational expressions are one of the types of expressions studied in algebra lessons at school. Let's give a definition.

Definition.

Expressions made up of numbers, variables, brackets, degrees with integer exponents, connected using the signs of arithmetic operations +, −, · and:, where division can be indicated by a fraction bar, are called rational expressions.

Here are some examples of rational expressions: .

Rational expressions begin to be purposefully studied in the 7th grade. Moreover, in the 7th grade, the basics of working with the so-called whole rational expressions, that is, with rational expressions that do not contain division into expressions with variables. To do this, monomials and polynomials are consistently studied, as well as the principles for performing actions with them. All this knowledge eventually allows you to perform the transformation of integer expressions.

In grade 8, they move on to the study of rational expressions containing division by an expression with variables, which are called fractional rational expressions. At the same time, special attention is paid to the so-called rational fractions(also called algebraic fractions), that is, fractions whose numerator and denominator contain polynomials. This ultimately makes it possible to perform the transformation of rational fractions.

The acquired skills allow us to proceed to the transformation of rational expressions of an arbitrary form. This is explained by the fact that any rational expression can be considered as an expression composed of rational fractions and integer expressions, connected by signs of arithmetic operations. And we already know how to work with integer expressions and algebraic fractions.

The main types of transformations of rational expressions

With rational expressions, you can carry out any of the basic identity transformations, whether it is a grouping of terms or factors, bringing similar terms, performing operations with numbers, etc. Typically, the purpose of these transformations is rational expression simplification.

Example.

Decision.

It is clear that this rational expression is the difference of two expressions and , moreover, these expressions are similar, since they have the same literal part. Thus, we can perform a reduction of like terms:

Answer:

It is clear that when carrying out transformations with rational expressions, as, indeed, with any other expressions, one must remain within the framework of the accepted order of actions.

Example.

Transform rational expression .

Decision.

We know that the actions in parentheses are executed first. Therefore, first of all, we transform the expression in brackets: 3 x − x=2 x .

Now you can substitute the result in the original rational expression: . So we came to an expression containing the actions of one stage - addition and multiplication.

Let's get rid of the parentheses at the end of the expression by applying the division-by-product property: .

Finally, we can group numeric factors and factors with variable x, and then perform the corresponding operations on numbers and apply : .

This completes the transformation of the rational expression, and as a result we got a monomial.

Answer:

Example.

Transform Rational Expression .

Decision.

First we convert the numerator and denominator. This order of transformation of fractions is explained by the fact that the stroke of a fraction is, in essence, another division designation, and the original rational expression is essentially a particular form , and the actions in parentheses are executed first.

So, in the numerator we perform operations with polynomials, first multiplication, then subtraction, and in the denominator we group the numerical factors and calculate their product: .

Let's also imagine the numerator and denominator of the resulting fraction as a product: suddenly it is possible to reduce the algebraic fraction. To do this, in the numerator we use difference of squares formula, and in the denominator we take the deuce out of brackets, we have .

Answer:

So, the initial acquaintance with the transformation of rational expressions can be considered accomplished. We pass, so to speak, to the sweetest.

Representation as a rational fraction

The most common end goal of transforming expressions is to simplify their form. In this light, the simplest form to which a fractionally rational expression can be converted is a rational (algebraic) fraction, and in a particular case, a polynomial, a monomial, or a number.

Is it possible to represent any rational expression as a rational fraction? The answer is yes. Let's explain why this is so.

As we have already said, any rational expression can be considered as polynomials and rational fractions connected by plus, minus signs, multiply and divide. All relevant operations on polynomials yield a polynomial or a rational fraction. In turn, any polynomial can be converted into an algebraic fraction by writing it with a denominator 1. And addition, subtraction, multiplication and division of rational fractions as a result give a new rational fraction. Therefore, after performing all the operations with polynomials and rational fractions in a rational expression, we get a rational fraction.

Example.

Express as a rational fraction the expression .

Decision.

The original rational expression is the difference between a fraction and a product of fractions of the form . According to the order of operations, we must first perform the multiplication, and only then the addition.

We start by multiplying algebraic fractions:

We substitute the result obtained into the original rational expression: .

We have come to the subtraction of algebraic fractions with different denominators:

So, having performed actions with rational fractions that make up the original rational expression, we presented it as a rational fraction.

Answer:

To consolidate the material, we will analyze the solution of another example.

Example.

Express a rational expression as a rational fraction.