This lesson covers the concept of an algebraic fraction. People encounter fractions in the simplest life situations: when it is necessary to divide an object into several parts, for example, to cut a cake equally into ten people. Obviously, everyone gets a piece of the cake. In this case, we are faced with the concept of a numerical fraction, but a situation is possible when an object is divided into an unknown number of parts, for example, by x. In this case, the concept of a fractional expression arises. You have already become acquainted with whole expressions (not containing division into expressions with variables) and their properties in 7th grade. Next we will look at the concept of a rational fraction, as well as acceptable values of variables.
Subject:Algebraic fractions. Arithmetic operations on algebraic fractions
Lesson:Basic Concepts
1. Definition and examples of algebraic fractions
Rational expressions are divided into integer and fractional expressions.
Definition. Rational fraction is a fractional expression of the form , where are polynomials. - numerator denominator.
Examples rational expressions:
- fractional expressions; - whole expressions. In the first expression, for example, the numerator is , and the denominator is .
Meaning algebraic fraction like anyone algebraic expression, depends on the numerical value of the variables that are included in it. In particular, in the first example the value of the fraction depends on the values of the variables and , and in the second example only on the value of the variable .
2. Calculating the value of an algebraic fraction and two basic fraction problems
Let's consider the first typical task: calculating the value rational fraction for different values of the variables included in it.
Example 1. Calculate the value of the fraction for a) , b) , c)
Solution. Let's substitute the values of the variables into the indicated fraction: a) , b) , c) - does not exist (since you cannot divide by zero).
Answer: 3; 1; does not exist.
As you can see, two typical problems arise for any fraction: 1) calculating the fraction, 2) finding valid and invalid values letter variables.
Definition. Valid Variable Values- values of variables at which the expression makes sense. The set of all possible values of variables is called ODZ or domain.
3. Acceptable (ADV) and unacceptable values of variables in fractions with one variable
The value of literal variables may be invalid if the denominator of the fraction at these values is zero. In all other cases, the values of the variables are valid, since the fraction can be calculated.
Example 2. Establish at what values of the variable the fraction does not make sense.
Solution. For this expression to make sense, it is necessary and sufficient that the denominator of the fraction does not equal zero. Thus, only those values of the variable will be invalid for which the denominator is equal to zero. The denominator of the fraction is , so we solve the linear equation:
Therefore, given the value of the variable, the fraction has no meaning.
From the solution of the example, the rule for finding invalid values of variables follows - the denominator of the fraction is equal to zero and the roots of the corresponding equation are found.
Let's look at several similar examples.
Example 3. Establish at what values of the variable the fraction does not make sense.
Solution. .
Answer. .
Example 4. Establish at what values of the variable the fraction does not make sense.
Solution..
There are other formulations of this problem - find domain or range of acceptable expression values (APV). This means finding all valid variable values. In our example, these are all values except . It is convenient to depict the domain of definition on a number axis.
To do this, we will cut out a point on it, as indicated in the figure:
Thus, fraction definition domain there will be all numbers except 3.
Answer..
Example 5. Establish at what values of the variable the fraction does not make sense.
Solution..
Let us depict the resulting solution on the numerical axis:
Answer..
4. Graphical representation of the area of acceptable (AP) and unacceptable values of variables in fractions
Example 6. Establish at what values of the variables the fraction does not make sense.
Solution.. We have obtained the equality of two variables, we will give numerical examples: or, etc.
Let us depict this solution on a graph in the Cartesian coordinate system:
Rice. 3. Function graph.
The coordinates of any point lying on this graph are not included in the range of acceptable fraction values.
Answer. .
5. Case of "division by zero" type
In the examples discussed, we encountered a situation where division by zero occurred. Now consider the case where a more interesting situation arises with type division.
Example 7. Establish at what values of the variables the fraction does not make sense.
Solution..
It turns out that the fraction makes no sense at . But one could argue that this is not the case because: 
.
It may seem that if the final expression is equal to 8 at , then the original one can also be calculated, and therefore makes sense at . However, if we substitute it into the original expression, we get - it makes no sense.
Answer..
To understand this example in more detail, let’s solve the following problem: at what values does the indicated fraction equal zero?
(a fraction is zero when its numerator is zero) 
. But it is necessary to solve the original equation with a fraction, and it does not make sense for , since at this value of the variable the denominator is zero. This means that this equation has only one root.
6. Rule for finding ODZ
Thus, we can formulate an exact rule for finding the range of permissible values of a fraction: to find ODZfractions it is necessary and sufficient to equate its denominator to zero and find the roots of the resulting equation.
We considered two main tasks: calculating the value of a fraction for the specified values of the variables and finding the range of acceptable values of a fraction.
Let's now consider a few more problems that may arise when working with fractions.
7. Various tasks and conclusions
Example 8. Prove that for any values of the variable the fraction .
Proof. The numerator is a positive number. . As a result, both the numerator and the denominator are positive numbers, therefore the fraction is a positive number.
Proven.
Example 9. It is known that , find .
Solution. Let's divide the fraction term by term. We have the right to reduce by, taking into account the fact that this is an invalid variable value for a given fraction.
Answer..
In this lesson we covered basic concepts related to fractions. In the next lesson we will look at main property of a fraction.
Bibliography
1. Bashmakov M.I. Algebra 8th grade. - M.: Education, 2004.
2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 8. - 5th ed. - M.: Education, 2010.
3. Nikolsky S. M., Potapov M. A., Reshetnikov N. N., Shevkin A. V. Algebra 8th grade. Textbook for general education institutions. - M.: Education, 2006.
1. Festival of pedagogical ideas.
2. Old school.
3. Internet portal lib2.podelise. ru.
Homework
1. No. 4, 7, 9, 12, 13, 14. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 8. - 5th ed. - M.: Education, 2010.
2. Write down a rational fraction whose domain of definition is: a) the set, b) the set, c) the entire number line.
3. Prove that for all possible values of the variable, the value of the fraction is non-negative.
4. Find the domain of expression. Instructions: consider separately two cases: when the denominator of the lower fraction is zero and when the denominator of the original fraction is zero.
In § 42 it was said that if the division of polynomials cannot be performed completely, then the quotient is written in the form of a fractional expression in which the dividend is the numerator and the divisor is the denominator.
Examples of fractional expressions:
The numerator and denominator of a fractional expression can themselves be fractional expressions, for example:
Of the fractional algebraic expressions, you most often have to deal with those in which the numerator and denominator are polynomials (in particular, monomials). Each such expression is called an algebraic fraction.
Definition. An algebraic expression that is a fraction whose numerator and denominator are polynomials is called an algebraic fraction.
As in arithmetic, the numerator and denominator of an algebraic fraction are called the terms of the fraction.
In the future, having studied operations on algebraic fractions, we will be able to transform any fractional expression into an algebraic fraction using identical transformations.
Examples of algebraic fractions:
Note that the whole expression, that is, a polynomial, can be written as a fraction; to do this, it is enough to write this expression in the numerator and 1 in the denominator. For example:
2. Acceptable letter values.
Letters included only in the numerator can take on any values (unless any additional restrictions are introduced by the condition of the problem).
For letters included in the denominator, only those values that do not turn the denominator to zero are valid. Therefore, in what follows we will always assume that the denominator of an algebraic fraction is not equal to zero.
When a student enters high school, mathematics is divided into two subjects: algebra and geometry. There are more and more concepts, the tasks are more and more difficult. Some people have difficulty understanding fractions. Missed the first lesson on this topic, and voila. fractions? A question that will torment throughout my school life.
The concept of an algebraic fraction
Let's start with a definition. Under algebraic fraction refers to the expressions P/Q, where P is the numerator and Q is the denominator. A number, a numerical expression, or a numerical-alphabetic expression can be hidden under a letter entry.
Before wondering how to solve algebraic fractions, you first need to understand that such an expression is part of the whole.
As a rule, an integer is 1. The number in the denominator shows how many parts the unit is divided into. The numerator is needed to find out how many elements are taken. The fraction bar corresponds to the division sign. It is allowed to write a fractional expression as a mathematical operation “Division”. In this case, the numerator is the dividend, the denominator is the divisor.
Basic rule of common fractions
When students study this topic at school, they are given examples to reinforce. To solve them correctly and find different ways out of complex situations, you need to apply the basic property of fractions.
It goes like this: If you multiply both the numerator and the denominator by the same number or expression (other than zero), the value of the common fraction does not change. A special case of this rule is the division of both sides of an expression by the same number or polynomial. Such transformations are called identical equalities.
Below we will look at how to solve addition and subtraction of algebraic fractions, multiplying, dividing and reducing fractions.
Mathematical operations with fractions
Let's look at how to solve, the main property of an algebraic fraction, and how to apply it in practice. If you need to multiply two fractions, add them, divide one by another, or subtract, you must always follow the rules.
Thus, for the operation of addition and subtraction, an additional factor must be found in order to bring the expressions to a common denominator. If the fractions are initially given with the same expressions Q, then this paragraph should be omitted. Once the common denominator is found, how do you solve algebraic fractions? You need to add or subtract numerators. But! It must be remembered that if there is a “-” sign in front of the fraction, all signs in the numerator are reversed. Sometimes you should not perform any substitutions or mathematical operations. It is enough to change the sign in front of the fraction.
The concept is often used as reducing fractions. This means the following: if the numerator and denominator are divided by an expression different from one (the same for both parts), then a new fraction is obtained. The dividend and divisor are smaller than before, but due to the basic rule of fractions they remain equal to the original example.
The purpose of this operation is to obtain a new irreducible expression. You can solve this problem by reducing the numerator and denominator by the greatest common factor. The operation algorithm consists of two points:
- Finding gcd for both sides of the fraction.
- Dividing the numerator and denominator by the found expression and obtaining an irreducible fraction equal to the previous one.
Below is a table showing the formulas. For convenience, you can print it out and carry it with you in a notebook. However, so that in the future, when solving a test or exam, there will be no difficulties in the question of how to solve algebraic fractions, these formulas must be learned by heart.
Several examples with solutions
From a theoretical point of view, the question of how to solve algebraic fractions is considered. The examples given in the article will help you better understand the material.
1. Convert fractions and bring them to a common denominator.
2. Convert fractions and bring them to a common denominator.
After studying the theoretical part and considering the practical part, no more questions should arise.
This lesson covers the concept of an algebraic fraction. People encounter fractions in the simplest life situations: when it is necessary to divide an object into several parts, for example, to cut a cake equally into ten people. Obviously, everyone gets a piece of the cake. In this case, we are faced with the concept of a numerical fraction, but a situation is possible when an object is divided into an unknown number of parts, for example, by x. In this case, the concept of a fractional expression arises. You have already become acquainted with whole expressions (not containing division into expressions with variables) and their properties in 7th grade. Next we will look at the concept of a rational fraction, as well as acceptable values of variables.
Rational expressions are divided into integer and fractional expressions.
Definition.Rational fraction is a fractional expression of the form , where are polynomials. - numerator denominator.
Examplesrational expressions:
- fractional expressions; - whole expressions. In the first expression, for example, the numerator is , and the denominator is .
Meaning algebraic fraction like anyone algebraic expression, depends on the numerical value of the variables that are included in it. In particular, in the first example the value of the fraction depends on the values of the variables and , and in the second example only on the value of the variable .
Let's consider the first typical task: calculating the value rational fraction for different values of the variables included in it.
Example 1. Calculate the value of the fraction for a) , b) , c)
Solution. Let's substitute the values of the variables into the indicated fraction: a) , b) , c) - does not exist (since you cannot divide by zero).
Answer: a) 3; b) 1; c) does not exist.
As you can see, two typical problems arise for any fraction: 1) calculating the fraction, 2) finding valid and invalid values letter variables.
Definition.Valid Variable Values- values of variables at which the expression makes sense. The set of all possible values of variables is called ODZ or domain.
The value of literal variables may be invalid if the denominator of the fraction at these values is zero. In all other cases, the values of the variables are valid, since the fraction can be calculated.
Example 2.
Solution. For this expression to make sense, it is necessary and sufficient that the denominator of the fraction does not equal zero. Thus, only those values of the variable will be invalid for which the denominator is equal to zero. The denominator of the fraction is , so we solve the linear equation:
Therefore, given the value of the variable, the fraction has no meaning.
Answer: -5.
From the solution of the example, the rule for finding invalid values of variables follows - the denominator of the fraction is equal to zero and the roots of the corresponding equation are found.
Let's look at several similar examples.
Example 3. Establish at what values of the variable the fraction does not make sense .
Solution..
Answer..
Example 4. Establish at what values of the variable the fraction does not make sense.
Solution..
There are other formulations of this problem - find domain or range of acceptable expression values (APV). This means finding all valid variable values. In our example, these are all values except . It is convenient to depict the domain of definition on a number axis.
To do this, we will cut out a point on it, as indicated in the figure:
Rice. 1
Thus, fraction definition domain there will be all numbers except 3.
Answer..
Example 5. Establish at what values of the variable the fraction does not make sense.
Solution..
Let us depict the resulting solution on the numerical axis:
Rice. 2
Answer..
Example 6.
Solution.. We have obtained the equality of two variables, we will give numerical examples: or, etc.
Let us depict this solution on a graph in the Cartesian coordinate system:
Rice. 3. Graph of a function
The coordinates of any point lying on this graph are not included in the range of acceptable fraction values.
Answer..
In the examples discussed, we encountered a situation where division by zero occurred. Now consider the case where a more interesting situation arises with type division.
Example 7. Establish at what values of the variables the fraction does not make sense.
Solution..
It turns out that the fraction makes no sense at . But one could argue that this is not the case because: 
.
It may seem that if the final expression is equal to 8 at , then the original one can also be calculated, and therefore makes sense at . However, if we substitute it into the original expression, we get - it makes no sense.
Answer..
To understand this example in more detail, let’s solve the following problem: at what values does the indicated fraction equal zero?
§ 1 The concept of an algebraic fraction
An algebraic fraction is the expression
where P and Q are polynomials; P is the numerator of the algebraic fraction, Q is the denominator of the algebraic fraction.
Here are examples of algebraic fractions:
Any polynomial is a special case of an algebraic fraction, because any polynomial can be written in the form
For example:
The value of an algebraic fraction depends on the value of the variables.
For example, let's calculate the value of the fraction
1)
2)
In the first case we get:
Note that this fraction can be reduced:
Thus, calculating the value of an algebraic fraction is simplified. Let's take advantage of this.
In the second case we get:
As you can see, with the change in the values of the variables, the value of the algebraic fraction changed.
§ 2 Admissible values of variables of an algebraic fraction
Consider the algebraic fraction
The value x = -1 is invalid for this fraction, because the denominator of the fraction at this value of x becomes zero. With this value of the variable, the algebraic fraction has no meaning.
Thus, the permissible values of the variables of an algebraic fraction are those values of the variables at which the denominator of the fraction does not vanish.
Let's solve a few examples.
At what values of the variable does the algebraic fraction make no sense:
To find invalid values of variables, the denominator of the fraction is set to zero, and the roots of the corresponding equation are found.
At what values of the variable is the algebraic fraction equal to zero:
A fraction is equal to zero if the numerator is zero. Let’s equate the numerator of our fraction to zero and find the roots of the resulting equation:
Thus, for x = 0 and x = 3, this algebraic fraction does not make sense, which means we must exclude these values of the variable from the answer.
So, in this lesson you learned the basic concepts of an algebraic fraction: the numerator and denominator of a fraction, as well as the acceptable values of the variables of an algebraic fraction.
List of used literature:
- Mordkovich A.G. "Algebra" 8th grade. At 2 hours. Part 1 Textbook for educational institutions / A.G. Mordkovich. – 9th ed., revised. – M.: Mnemosyne, 2007. – 215 p.: ill.
- Mordkovich A.G. "Algebra" 8th grade. At 2 hours. Part 2 Problem book for educational institutions / A.G. Mordkovich, T.N. Mishustina, E.E. Tulchinskaya. – 8th ed., – M.: Mnemosyne, 2006 – 239 p.
- Algebra. 8th grade. Tests for students of educational institutions of L.A. Alexandrov, ed. A.G. Mordkovich 2nd ed., erased. - M.: Mnemosyne 2009. - 40 p.
- Algebra. 8th grade. Independent work for students of educational institutions: to the textbook by A.G. Mordkovich, L.A. Alexandrov, ed. A.G. Mordkovich. 9th ed., erased. - M.: Mnemosyne 2013. - 112 p.
