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 values of the variables. 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?
48. Types of algebraic expressions.
Algebraic expressions are constructed from numbers and variables using the signs of addition, subtraction, multiplication, division, raising to a rational power and extracting roots and using parentheses.
Examples of algebraic expressions:
If an algebraic expression does not contain division into variables and extraction of roots from variables (in particular, exponentiation with a fractional exponent), then it is called an integer. Of the ones written above, expressions 1, 2 and 6 are integers.
If an algebraic expression is composed of numbers and variables using the operations of addition, subtraction, multiplication, exponentiation with a natural exponent and division, and division into expressions with variables is used, then it is called fractional. So, of those written above, expressions 3 and 4 are fractional.
Integer and fractional expressions are called rational expressions. So, of the rational expressions written above, expressions 1, 2, 3, 4 and 6 are.
If an algebraic expression involves taking the root of variables (or raising variables to a fractional power), then such an algebraic expression is called irrational. Thus, of those written above, expressions 5 and 7 are irrational.
So, algebraic expressions can be rational and irrational. Rational expressions, in turn, are divided into integers and fractions.
49. Valid values of variables. The domain of definition of an algebraic expression.
The values of the variables for which the algebraic expression makes sense are called admissible values of the variables. The set of all permissible values of variables is called the domain of definition of an algebraic expression.
The whole expression makes sense for any values of the variables included in it. So, for any values of the variables, the whole expressions 1, 2, 6 from paragraph 48 make sense.
Fractional expressions do not make sense for those values of the variables that make the denominator zero. Thus, fractional expression 3 from paragraph 48 makes sense for all o, except , and fractional expression 4 makes sense for all a, b, c, except for the values of a
The irrational expression does not make sense for those values of the variables that turn into a negative number the expression contained under the sign of the root of an even power or under the sign of raising to a fractional power. Thus, the irrational expression 5 makes sense only for those a, b for which and the irrational expression 7 makes sense only for and (see paragraph 48).
If in an algebraic expression the variables are given valid values, then a numerical expression will be obtained; its value is called the value of the algebraic expression for the selected values of the variables.
Example. Find the value of the expression when
Solution. We have
50. The concept of identical transformation of an expression. Identity.
Let's consider two expressions When we have . The numbers 0 and 3 are called their respective values. expressions for Let us find the corresponding values of the same expressions for
The corresponding values of two expressions can be equal to each other (for example, in the considered example, equality is true), or they can differ from each other (for example, in the considered example).
Valid values of variables,
included in a fractional expression
Goals: develop the ability to find acceptable values of variables included in fractional expressions.
During the classes
I. Organizational moment.
II. Oral work.
– Substitute some number instead of * and name the resulting fraction:
A) ; b) ; V) ; G) ;
d) ; e) ; and) ; h) .
III. Explanation of new material.
The explanation of new material occurs in three stages:
1. Updating students' knowledge.
2. Consideration of the question of whether a rational fraction always makes sense.
3. Derivation of the rule for finding acceptable values of variables included in a rational fraction.
When updating knowledge, students can be asked the following:
questions:
– What fraction is called rational?
– Is every fraction a fractional expression?
– How to find the value of a rational fraction for given values of the variables included in it?
To clarify the issue of acceptable values of the variables included in a rational fraction, you can ask students to complete a task.
Assignment: Find the value of the fraction for the specified values of the variable:
At X = 4; 0; 1.
By completing this task, students understand that when X= 1 it is impossible to find the value of the fraction. This allows them to make the following conclusion: you cannot substitute numbers into a rational fraction that make its denominator zero (this conclusion must be formulated and spoken out loud by the students themselves).
After this, the teacher informs the students that all values of the variables for which the rational expression makes sense are called valid values of the variables.
1) If the expression is an integer, then all the values of the variables included in it will be valid.
2) To find acceptable values of the variables of a fractional expression, you need to check at what values the denominator goes to zero. The numbers found will not be valid values.
IV. Formation of skills and abilities.
1. № 10, № 11.
The answer to the question about the acceptable values of the variables included in a fractional expression may sound different. For example, when considering a rational fraction, we can say that all numbers except X= 4, or that the permissible values of the variable do not include the number 4, that is X ≠ 4.
Both formulations are correct; the main thing is to ensure that the format is correct.
SAMPLE FORM:
4X (X + 1) = 0
Answer: X≠ 0 and X≠ 1 (or all numbers except 0 and –1).
3. No. 14 (a, c), No. 15.
When completing these tasks, students should pay attention to the need to take into account the permissible values of variables.
G) 
Answer: X = 0.
Monitor the rationale for all reasoning.
In a class with a high level of training, you can additionally perform No. 18 and No. 20.
Solution
Of all fractions with the same positive numerator, the larger one is the one with the smallest denominator. That is, it is necessary to find at what value A expression A 2 + 5 takes the smallest value.
Since the expression A 2 cannot be negative for any value A, then the expression A 2 + 5 will take the smallest value when A = 0.
Answer: A = 0.
Arguing similarly, we find that it is necessary to find the value A, for which the expression ( A– 3) 2 + 1 takes the smallest value.
Answer: A = 3.
Solution
.
To answer the question, you first need to transform the expression in the denominator of the fraction.
The fraction will take on the greatest value if the expression (2 X +
+ at) 2 + 9 takes the smallest value. Since (2 X + at) 2 cannot take negative values, then the smallest value of the expression (2 X + at) 2 + 9 equals 9.
Then the value of the original fraction is = 2.
V. Lesson summary.
Frequently asked questions:
– What values are called acceptable values of the variables included in the expression?
– What are the valid values for the variables of an entire expression?
– How to find valid values for variables in a fractional expression?
– Are there rational fractions for which all variable values are valid? Give examples of such fractions.
Homework: No. 12, No. 14 (b, d), No. 212.