A rational expression is an algebraic expression that does not contain radicals. In other words, this is one or more algebraic quantities (numbers and letters) interconnected by signs of arithmetic operations: addition, subtraction, multiplication ... ... Wikipedia
An algebraic expression that does not contain radicals and includes only the operations of addition, subtraction, multiplication, and division. For example, a2 + b, x/(y z2) … Big Encyclopedic Dictionary
An algebraic expression that does not contain radicals and includes only the operations of addition, subtraction, multiplication, and division. For example, a2 + b, x/(y z2). * * * RATIONAL EXPRESSION RATIONAL EXPRESSION, an algebraic expression that does not contain ... ... encyclopedic Dictionary
An algebraic expression that does not contain radicals, such as a2 + b, x/(y z3). If included in R. century. letters are considered variables, then R. in. defines a rational function (See Rational function) of these variables ... Great Soviet Encyclopedia
An algebraic expression that does not contain radicals and includes only the operations of addition, subtraction, multiplication, and division. For example, a2 + b, x/(y z2) ... Natural science. encyclopedic Dictionary
EXPRESSION- the primary mathematical concept, which means a record of letters and numbers connected by signs of arithmetic operations, while brackets, function designations, etc. can be used; usually B is the formula million part of it. Distinguish In (1) ... ... Great Polytechnic Encyclopedia
RATIONAL- (Rational; Rational) a term used to describe thoughts, feelings and actions consistent with the mind; an attitude based on objective values obtained as a result of practical experience. “Objective values are established in experience ... ... Analytical Psychology Dictionary
RATIONAL KNOWLEDGE- a subjective image of the objective world, obtained with the help of thinking. Thinking is an active process of generalized and indirect reflection of reality, which ensures the discovery of its regular connections on the basis of sensory data and their expression ... Philosophy of Science and Technology: Thematic Dictionary
EQUATION, RATIONAL- A logical or mathematical expression based on (rational) assumptions about processes. Such equations differ from empirical equations in that their parameters are obtained as a result of deductive conclusions from theoretical ... ... Explanatory Dictionary of Psychology
RATIONAL, rational, rational; rational, rational, rational. 1. adj. to rationalism (book). rational philosophy. 2. Quite reasonable, justified, expedient. He made a rational suggestion. Rational ... ... Explanatory Dictionary of Ushakov
1) R. algebraic equation f(x)=0 of degree p algebraic equation g(y)=0 with coefficients rationally dependent on the coefficients f (x), such that knowledge of the roots of this equation allows us to find the roots of this equation ... ... Mathematical Encyclopedia
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.
Page navigation.
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. 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.
An integer expression is a mathematical expression made up of numbers and literal variables using the operations of addition, subtraction, and multiplication. Integers also include expressions that include division by some number other than zero.
Integer Expression Examples
Below are some examples of integer expressions:
1. 12*a^3 + 5*(2*a -1);
3. 4*y- ((5*y+3)/5) -1;
Fractional Expressions
If the expression contains a division by a variable or by another expression containing a variable, then such an expression is not an integer. Such an expression is called a fractional expression. Let us give a complete definition of a fractional expression.
A fractional expression is a mathematical expression that, in addition to the operations of addition, subtraction and multiplication performed with numbers and literal variables, as well as division by a number not equal to zero, also contains division into expressions with literal variables.
Examples of fractional expressions:
1. (12*a^3 +4)/a
3. 4*x- ((5*y+3)/(5-y)) +1;
Fractional and integer expressions make up two large sets of mathematical expressions. If these sets are combined, then we get a new set, which is called rational expressions. That is, rational expressions are all integer and fractional expressions.
We know that integer expressions make sense for any values of the variables that are included in it. This follows from the fact that in order to find the value of an integer expression, it is necessary to perform actions that are always possible: addition, subtraction, multiplication, division by a number other than zero.
Fractional expressions, unlike integer ones, may not make sense. Since there is a division operation by a variable or an expression containing variables, and this expression can turn to zero, but division by zero is impossible. Variable values for which the fractional expression will make sense are called valid variable values.
rational fraction
One of the special cases of rational expressions will be a fraction, the numerator and denominator of which are polynomials. For such a fraction in mathematics, there is also a name - a rational fraction.
A rational fraction will make sense if its denominator is not equal to zero. That is, all values of variables for which the denominator of the fraction is different from zero will be valid.
Important notes!
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2. Before you start reading the article, pay attention to our navigator for the most useful resource for
Often we hear this unpleasant phrase: "simplify the expression." Usually, in this case, we have some kind of monster like this:
“Yes, much easier,” we say, but such an answer usually does not work.
Now I will teach you not to be afraid of any such tasks.
Moreover, at the end of the lesson, you yourself will simplify this example to a (just!) ordinary number (yes, to hell with these letters).
But before you start this lesson, you need to be able to deal with fractions and factorize polynomials.
Therefore, if you have not done this before, be sure to master the topics "" and "".
Read? If yes, then you are ready.
Let's go! (Let's go!)
Basic Expression Simplification Operations
Now we will analyze the main techniques that are used to simplify expressions.
The simplest of them is
1. Bringing similar
What are similar? You went through this in 7th grade, when letters first appeared in math instead of numbers.
Similar are terms (monomials) with the same letter part.
For example, in the sum, like terms are and.
Remembered?
Bring similar- means to add several similar terms with each other and get one term.
But how can we put letters together? - you ask.
This is very easy to understand if you imagine that the letters are some kind of objects.
For example, the letter is a chair. Then what is the expression?
Two chairs plus three chairs, how much will it be? That's right, chairs: .
Now try this expression:
In order not to get confused, let different letters denote different objects.
For example, - this is (as usual) a chair, and - this is a table.
chairs tables chair tables chairs chairs tables
The numbers by which the letters in such terms are multiplied are called coefficients.
For example, in the monomial the coefficient is equal. And he is equal.
So, the rule for bringing similar:
Examples:
Bring similar:
Answers:
2. (and are similar, since, therefore, these terms have the same letter part).
2. Factorization
This is usually the most important part in simplifying expressions.
After you have given similar ones, most often the resulting expression is needed factorize, i.e. represent as a product.
Especially this important in fractions: because in order to reduce the fraction, the numerator and denominator must be expressed as a product.
You went through the detailed methods of factoring expressions in the topic "", so here you just have to remember what you have learned.
To do this, solve a few examples (you need to factorize)
Examples:
Solutions:
3. Fraction reduction.
Well, what could be nicer than to cross out part of the numerator and denominator, and throw them out of your life?
That's the beauty of abbreviation.
It's simple:
If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.
This rule follows from the basic property of a fraction:
That is, the essence of the reduction operation is that We divide the numerator and denominator of a fraction by the same number (or by the same expression).
To reduce a fraction, you need:
1) numerator and denominator factorize
2) if the numerator and denominator contain common factors, they can be deleted.
Examples:
The principle, I think, is clear?
I would like to draw your attention to one typical mistake in abbreviation. Although this topic is simple, but many people do everything wrong, not realizing that cut- it means divide numerator and denominator by the same number.
No abbreviations if the numerator or denominator is the sum.
For example: you need to simplify.
Some do this: which is absolutely wrong.
Another example: reduce.
The "smartest" will do this:
Tell me what's wrong here? It would seem: - this is a multiplier, so you can reduce.
But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not decomposed into factors.
Here is another example: .
This expression is decomposed into factors, which means that you can reduce, that is, divide the numerator and denominator by, and then by:
You can immediately divide by:
To avoid such mistakes, remember an easy way to determine if an expression is factored:
The arithmetic operation that is performed last when calculating the value of the expression is the "main".
That is, if you substitute some (any) numbers instead of letters, and try to calculate the value of the expression, then if the last action is multiplication, then we have a product (the expression is decomposed into factors).
If the last action is addition or subtraction, this means that the expression is not factored (and therefore cannot be reduced).
To fix it yourself, a few examples:
Examples:
Solutions:
4. Addition and subtraction of fractions. Bringing fractions to a common denominator.
Addition and subtraction of ordinary fractions is a well-known operation: we look for a common denominator, multiply each fraction by the missing factor and add / subtract the numerators.
Let's remember:
Answers:
1. The denominators and are coprime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:
2. Here the common denominator is:
3. Here, first of all, we turn mixed fractions into improper ones, and then - according to the usual scheme:
It is quite another matter if the fractions contain letters, for example:
Let's start simple:
a) Denominators do not contain letters
Here everything is the same as with ordinary numerical fractions: we find a common denominator, multiply each fraction by the missing factor and add / subtract the numerators:
now in the numerator you can bring similar ones, if any, and factor them:
Try it yourself:
Answers:
b) Denominators contain letters
Let's remember the principle of finding a common denominator without letters:
First of all, we determine the common factors;
Then we write out all the common factors once;
and multiply them by all other factors, not common ones.
To determine the common factors of the denominators, we first decompose them into simple factors:
We emphasize the common factors:
Now we write out the common factors once and add to them all non-common (not underlined) factors:
This is the common denominator.
Let's get back to the letters. The denominators are given in exactly the same way:
We decompose the denominators into factors;
determine common (identical) multipliers;
write out all the common factors once;
We multiply them by all other factors, not common ones.
So, in order:
1) decompose the denominators into factors:
2) determine the common (identical) factors:
3) write out all the common factors once and multiply them by all the other (not underlined) factors:
So the common denominator is here. The first fraction must be multiplied by, the second - by:
By the way, there is one trick:
For example: .
We see the same factors in the denominators, only all with different indicators. The common denominator will be:
to the extent
to the extent
to the extent
in degree.
Let's complicate the task:
How to make fractions have the same denominator?
Let's remember the basic property of a fraction:
Nowhere is it said that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!
See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What has been learned?
So, another unshakable rule:
When you bring fractions to a common denominator, use only the multiplication operation!
But what do you need to multiply to get?
Here on and multiply. And multiply by:
Expressions that cannot be factorized will be called "elementary factors".
For example, is an elementary factor. - too. But - no: it is decomposed into factors.
What about expression? Is it elementary?
No, because it can be factorized:
(you already read about factorization in the topic "").
So, the elementary factors into which you decompose an expression with letters are an analogue of the simple factors into which you decompose numbers. And we will do the same with them.
We see that both denominators have a factor. It will go to the common denominator in the power (remember why?).
The multiplier is elementary, and they do not have it in common, which means that the first fraction will simply have to be multiplied by it:
Another example:
Decision:
Before multiplying these denominators in a panic, you need to think about how to factor them? Both of them represent:
Fine! Then:
Another example:
Decision:
As usual, we factorize the denominators. In the first denominator, we simply put it out of brackets; in the second - the difference of squares:
It would seem that there are no common factors. But if you look closely, they are already so similar ... And the truth is:
So let's write:
That is, it turned out like this: inside the bracket, we swapped the terms, and at the same time, the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.
Now we bring to a common denominator:
Got it? Now let's check.
Tasks for independent solution:
Answers:
5. Multiplication and division of fractions.
Well, the hardest part is now over. And ahead of us is the simplest, but at the same time the most important:
Procedure
What is the procedure for calculating a numeric expression? Remember, considering the value of such an expression:
Did you count?
It should work.
So, I remind you.
The first step is to calculate the degree.
The second is multiplication and division. If there are several multiplications and divisions at the same time, you can do them in any order.
And finally, we perform addition and subtraction. Again, in any order.
But: the parenthesized expression is evaluated out of order!
If several brackets are multiplied or divided by each other, we first evaluate the expression in each of the brackets, and then multiply or divide them.
What if there are other parentheses inside the brackets? Well, let's think: some expression is written inside the brackets. What is the first thing to do when evaluating an expression? That's right, calculate brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.
So, the order of actions for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):
Okay, it's all simple.
But that's not the same as an expression with letters, is it?
No, it's the same! Only instead of arithmetic operations it is necessary to do algebraic operations, that is, the operations described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use it when working with fractions). Most often, for factorization, you need to use i or simply take the common factor out of brackets.
Usually our goal is to represent an expression as a product or quotient.
For example:
Let's simplify the expression.
1) First we simplify the expression in brackets. There we have the difference of fractions, and our goal is to represent it as a product or quotient. So, we bring the fractions to a common denominator and add:
It is impossible to simplify this expression further, all factors here are elementary (do you still remember what this means?).
2) We get:
Multiplication of fractions: what could be easier.
3) Now you can shorten:
That's it. Nothing complicated, right?
Another example:
Simplify the expression.
First, try to solve it yourself, and only then look at the solution.
Decision:
First of all, let's define the procedure.
First, let's add the fractions in brackets, instead of two fractions, one will turn out.
Then we will do the division of fractions. Well, we add the result with the last fraction.
I will schematically number the steps:
Finally, I will give you two useful tips:
1. If there are similar ones, they must be brought immediately. At whatever moment we have similar ones, it is advisable to bring them right away.
2. The same goes for reducing fractions: as soon as an opportunity arises to reduce, it must be used. The exception is fractions that you add or subtract: if they now have the same denominators, then the reduction should be left for later.
Here are some tasks for you to solve on your own:
And promised at the very beginning:
Answers:
Solutions (brief):
If you coped with at least the first three examples, then you, consider, have mastered the topic.
Now on to learning!
EXPRESSION CONVERSION. SUMMARY AND BASIC FORMULA
Basic simplification operations:
- Bringing similar: to add (reduce) like terms, you need to add their coefficients and assign the letter part.
- Factorization: taking the common factor out of brackets, applying, etc.
- Fraction reduction: the numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, from which the value of the fraction does not change.
1) numerator and denominator factorize
2) if there are common factors in the numerator and denominator, they can be crossed out.IMPORTANT: only multipliers can be reduced!
- Addition and subtraction of fractions:
; - Multiplication and division of fractions:
;
Well, the topic is over. If you are reading these lines, then you are very cool.
Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!
Now the most important thing.
You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough ...
For what?
For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.
I will not convince you of anything, I will just say one thing ...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the exam and be ultimately ... happier?
FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.
On the exam, you will not be asked theory.
You will need solve problems on time.
And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.
It's like in sports - you need to repeat many times to win for sure.
Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!
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Find problems and solve!
This lesson will cover the basic information about rational expressions and their transformations, as well as examples of the transformation of rational expressions. This topic summarizes the topics we have studied so far. Transformations of rational expressions include addition, subtraction, multiplication, division, raising to the power of algebraic fractions, reduction, factorization, etc. As part of the lesson, we will look at what a rational expression is, and also analyze examples for their transformation.
Subject:Algebraic fractions. Arithmetic operations on algebraic fractions
Lesson:Basic information about rational expressions and their transformations
Definition
rational expression is an expression consisting of numbers, variables, arithmetic operations and exponentiation.
Consider an example of a rational expression:
Special cases of rational expressions:
1st degree: 
;
2. monomial: ;
3. fraction: .
Rational Expression Transformation is a simplification of a rational expression. The order of operations when converting rational expressions: first, there are actions in brackets, then multiplication (division), and then addition (subtraction) operations.
Let's consider some examples on transformation of rational expressions.
Example 1
Decision:
Let's solve this example step by step. The action in parentheses is performed first.
Answer:
Example 2
Decision:
Answer:
Example 3
Decision:
Answer: .
Note: perhaps, at the sight of this example, an idea occurred to you: reduce the fraction before reducing to a common denominator. Indeed, it is absolutely correct: first, it is desirable to simplify the expression as much as possible, and then transform it. Let's try to solve the same example in the second way.
As you can see, the answer turned out to be absolutely similar, but the solution turned out to be somewhat simpler.
In this lesson, we looked at rational expressions and their transformations, as well as several specific examples of these transformations.
Bibliography
1. Bashmakov M.I. Algebra 8th grade. - M.: Enlightenment, 2004.
2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 8. - 5th ed. - M.: Education, 2010.
