Algebraic expressions 7. Methodological development in algebra on the topic: Algebraic expressions and their characteristics

Algebraic expressions begin to be studied in 7th grade. They have a number of properties and are used in solving problems. Let's study this topic in more detail and consider an example of solving the problem.

Definition of the concept

What expressions are called algebraic? This mathematical notation made up of numbers, letters and symbols arithmetic operations. The presence of letters is the main difference between numerical and algebraic expressions. Examples:

• 4a+5;
• 6b-8;
• 5s:6*(8+5).

A letter in algebraic expressions denotes a number. That's why it's called a variable - in the first example it's the letter a, in the second it's b, and in the third it's c. The algebraic expression itself is also called expression with variable.

Expression value

Meaning of algebraic expression is the number obtained as a result of performing all the arithmetic operations indicated in this expression. But to get it, the letters must be replaced with numbers. Therefore, in the examples they always indicate which number corresponds to the letter. Let's look at how to find the value of the expression 8a-14*(5-a) if a=3.

Let's substitute the number 3 for the letter a. We get the following entry: 8*3-14*(5-3).

As in numerical expressions, the solution of an algebraic expression is carried out according to the rules for performing arithmetic operations. Let's solve everything in order.

• 5-3=2.
• 8*3=24.
• 14*2=28.
• 24-28=-4.

Thus, the value of the expression 8a-14*(5-a) at a=3 is equal to -4.

The value of a variable is called valid if the expression makes sense with it, that is, it is possible to find its solution.

An example of a valid variable for the expression 5:2a is the number 1. Substituting it into the expression, we get 5:2*1=2.5.

The invalid variable for this expression is 0. If we substitute zero into the expression, we get 5:2*0, that is, 5:0. You can't divide by zero, which means the expression doesn't make sense.

Identity expressions

If two expressions are equal for any values ​​of their constituent variables, they are called identical.
Example of identical expressions :
4(a+c) and 4a+4c.
Whatever values ​​the letters a and c take, the expressions will always be equal. Any expression can be replaced by another that is identical to it. This process is called identity transformation.

Example of identity transformation .
4*(5a+14c) – this expression can be replaced by the identical one by applying mathematical law multiplication. To multiply a number by the sum of two numbers, you need to multiply this number by each term and add the results.

• 4*5a=20a.
• 4*14s=64s.
• 20a+64s.

Thus, the expression 4*(5a+14c) is identical to 20a+64c.

The number appearing before a letter variable in an algebraic expression is called a coefficient. The coefficient and the variable are multipliers.

Problem solving

Algebraic expressions are used to solve problems and equations.
Let's consider the problem. Petya came up with a number. In order for his classmate Sasha to guess it, Petya told him: first I added 7 to the number, then subtracted 5 from it and multiplied by 2. As a result, I got the number 28. What number did I guess?

To solve the problem, you need to denote the hidden number with the letter a, and then perform all specified actions with him.

• (a+7)-5.
• ((a+7)-5)*2=28.

Now let's solve the resulting equation.

Petya wished for the number 12.

What have we learned?

An algebraic expression is a record made up of letters, numbers and arithmetic symbols. Each expression has a value, which is found by performing all the arithmetic operations in the expression. The letter in an algebraic expression is called a variable, and the number in front of it is called a coefficient. Algebraic expressions are used to solve problems.

We can write some mathematical expressions in different ways. Depending on our goals, whether we have enough data, etc. Numerical and algebraic expressions They differ in that we write the first ones only as numbers combined using arithmetic symbols (addition, subtraction, multiplication, division) and parentheses.

If instead of numbers you enter Latin letters (variables) into the expression, it will become algebraic. Algebraic expressions use letters, numbers, addition and subtraction, multiplication and division signs. The sign of the root, degree, and parentheses can also be used.

In any case, whether the expression is numerical or algebraic, it cannot be just a random set of signs, numbers and letters - it must have meaning. This means that letters, numbers, signs must be connected by some kind of relationship. Correct example: 7x + 2: (y + 1). Bad example): + 7x - * 1.

The word “variable” was mentioned above - what does it mean? This is a Latin letter, instead of which you can substitute a number. And if we are talking about variables, in this case algebraic expressions can be called an algebraic function.

The variable can take different meanings. And by substituting some number in its place, we can find the value of the algebraic expression in this case specific meaning variable. When the value of a variable is different, the value of the expression will be different.

How to solve algebraic expressions?

To calculate the values ​​you need to do converting algebraic expressions. And for this you still need to take into account a few rules.

First: the scope of algebraic expressions is all possible values variables for which this expression can make sense. What is meant? For example, you cannot substitute a value for a variable that would require you to divide by zero. In the expression 1/(x – 2), 2 must be excluded from the domain of definition.

Secondly, remember how to simplify expressions: factor them, put identical variables out of brackets, etc. For example: if you swap the terms, the sum will not change (y + x = x + y). Likewise, the product will not change if the factors are swapped (x*y = y*x).

In general, they are excellent for simplifying algebraic expressions. abbreviated multiplication formulas. Those who have not yet learned them should definitely do so - they will still come in handy more than once:

we find the difference between the variables squared: x 2 – y 2 = (x – y)(x + y);

we find the sum squared: (x + y) 2 = x 2 + 2xy + y 2;

we calculate the difference squared: (x – y) 2 = x 2 – 2xy + y 2;

cube the sum: (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 or (x + y) 3 = x 3 + y 3 + 3xy (x + y);

cube the difference: (x – y) 3 = x 3 – 3x 2 y + 3xy 2 – y 3 or (x – y) 3 = x 3 – y 3 – 3xy (x – y);

we find the sum of the variables cubed: x 3 + y 3 = (x + y) (x 2 – xy + y 2);

we calculate the difference between the variables cubed: x 3 – y 3 = (x – y)(x 2 + xy + y 2);

we use the roots: xa 2 + ua + z = x(a – a 1)(a – a 2), and 1 and a 2 are the roots of the expression xa 2 + ua + z.

You should also have an understanding of the types of algebraic expressions. They are:

rational, and those in turn are divided into:

integers (there is no division into variables, no extraction of roots from variables and no raising to fractional powers): 3a 3 b + 4a 2 b * (a – b). The domain of definition is all possible values ​​of the variables;

fractional (except for others mathematical operations, such as addition, subtraction, multiplication, in these expressions they divide by a variable and raise to a power (with natural indicator): (2/b – 3/a + c/4) 2 . The domain of definition is all values ​​of the variables for which the expression is not equal to zero;

irrational - for an algebraic expression to be considered as such, it must contain raising variables to a power with fractional indicator and/or extracting roots from variables: √a + b 3/4. Domain of definition – all values ​​of variables, excluding those for which the expression under the root of an even degree or under fractional power becomes a negative number.

Identical transformations of algebraic expressions– one more useful trick to solve them. An identity is an expression that will be true for any variables included in the domain of definition that are substituted into it.

An expression that depends on some variables can be identically equal to another expression if it depends on the same variables and if the values ​​of both expressions are equal, no matter what values ​​of the variables are chosen. In other words, if an expression can be expressed in two different ways (expressions) whose meanings are the same, those expressions are identically equal. For example: y + y = 2y, or x 7 = x 4 * x 3, or x + y + z = z + x + y.

When performing tasks with algebraic expressions identity transformation serves so that one expression can be replaced by another that is identical to it. For example, replace x 9 with the product x 5 * x 4.

Examples of solutions

To make it clearer, let's look at a few examples. transformations of algebraic expressions. Tasks of this level can be found in KIMs for the Unified State Exam.

Task 1: Find the value of the expression ((12x) 2 – 12x)/(12x 2 -1).

Solution: ((12x) 2 – 12x)/(12x 2 – 1) = (12x (12x -1))/x*(12x – 1) = 12.

Task 2: Find the value of the expression (4x 2 – 9)*(1/(2x – 3) – 1/(2x +3).

Solution: (4x 2 – 9)*(1/(2x – 3) – 1/(2x +3) = (2x – 3)(2x + 3)(2x + 3 – 2x + 3)/(2x – 3 )(2x + 3) = 6.

Conclusion

When preparing for school tests, Unified State Examinations and GIA you can always use this material as a hint. Remember that an algebraic expression is a combination of numbers and variables expressed in Latin letters. And also signs of arithmetic operations (addition, subtraction, multiplication, division), parentheses, powers, roots.

Use abbreviated multiplication formulas and knowledge of identities to transform algebraic expressions.

Write us your comments and wishes in the comments - it is important for us to know that you are reading us.

blog.site, when copying material in full or in part, a link to the original source is required.

We can write some mathematical expressions in different ways. Depending on our goals, whether we have enough data, etc. Numeric and algebraic expressions They differ in that we write the first ones only as numbers combined using arithmetic symbols (addition, subtraction, multiplication, division) and parentheses.

If instead of numbers you enter Latin letters (variables) into the expression, it will become algebraic. Algebraic expressions use letters, numbers, addition and subtraction, multiplication and division signs. The sign of the root, degree, and parentheses can also be used.

In any case, whether the expression is numerical or algebraic, it cannot be just a random set of signs, numbers and letters - it must have meaning. This means that letters, numbers, signs must be connected by some kind of relationship. Correct example: 7x + 2: (y + 1). Bad example) : + 7x - * 1.

The word “variable” was mentioned above - what does it mean? This is a Latin letter, instead of which you can substitute a number. And if we are talking about variables, in this case algebraic expressions can be called an algebraic function.

The variable can take on different values. And by substituting some number in its place, we can find the value of the algebraic expression for this particular value of the variable. When the value of a variable is different, the value of the expression will be different.

How to solve algebraic expressions?

To calculate the values ​​you need to do converting algebraic expressions. And for this you still need to take into account a few rules.

First, the scope of algebraic expressions is all possible values ​​of a variable for which the expression can make sense. What is meant? For example, you cannot substitute a value for a variable that would require you to divide by zero. In the expression 1/(x – 2), 2 must be excluded from the domain of definition.

Secondly, remember how to simplify expressions: factor them, put identical variables out of brackets, etc. For example: if you swap the terms, the sum will not change (y + x = x + y). Likewise, the product will not change if the factors are swapped (x*y = y*x).

In general, they are excellent for simplifying algebraic expressions. abbreviated multiplication formulas. Those who have not yet learned them should definitely do so - they will still come in handy more than once:

we find the difference between the variables squared: x 2 – y 2 = (x – y)(x + y);

we find the sum squared: (x + y) 2 = x 2 + 2xy + y 2;

we calculate the difference squared: (x – y) 2 = x 2 – 2xy + y 2;

cube the sum: (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 or (x + y) 3 = x 3 + y 3 + 3xy (x + y);

cube the difference: (x – y) 3 = x 3 – 3x 2 y + 3xy 2 – y 3 or (x – y) 3 = x 3 – y 3 – 3xy (x – y);

we find the sum of the variables cubed: x 3 + y 3 = (x + y) (x 2 – xy + y 2);

we calculate the difference between the variables cubed: x 3 – y 3 = (x – y)(x 2 + xy + y 2);

we use the roots: xa 2 + ua + z = x(a – a 1)(a – a 2), and 1 and a 2 are the roots of the expression xa 2 + ua + z.

You should also have an understanding of the types of algebraic expressions. They are:

rational, and those in turn are divided into:

integers (there is no division into variables, no extraction of roots from variables and no raising to fractional powers): 3a 3 b + 4a 2 b * (a – b). The domain of definition is all possible values ​​of the variables;

fractional (except for other mathematical operations, such as addition, subtraction, multiplication, in these expressions they are divided by a variable and raised to a power (with a natural exponent): (2/b - 3/a + c/4) 2. Domain of definition - all values variables for which the expression is not equal to zero;

irrational - for an algebraic expression to be considered as such, it must involve raising variables to a power with a fractional exponent and/or extracting roots from variables: √a + b 3/4. The domain of definition is all values ​​of the variables, excluding those for which the expression under the root of an even power or under a fractional power becomes a negative number.

Identical transformations of algebraic expressions is another useful technique for solving them. An identity is an expression that will be true for any variables included in the domain of definition that are substituted into it.

An expression that depends on some variables can be identically equal to another expression if it depends on the same variables and if the values ​​of both expressions are equal, no matter what values ​​of the variables are chosen. In other words, if an expression can be expressed in two different ways (expressions) whose meanings are the same, those expressions are identically equal. For example: y + y = 2y, or x 7 = x 4 * x 3, or x + y + z = z + x + y.

When performing tasks with algebraic expressions, the identity transformation serves to ensure that one expression can be replaced by another that is identical to it. For example, replace x 9 with the product x 5 * x 4.

Examples of solutions

To make it clearer, let's look at a few examples. transformations of algebraic expressions. Tasks of this level can be found in KIMs for the Unified State Exam.

Task 1: Find the value of the expression ((12x) 2 – 12x)/(12x 2 -1).

Solution: ((12x) 2 – 12x)/(12x 2 – 1) = (12x (12x -1))/x*(12x – 1) = 12.

Task 2: Find the value of the expression (4x 2 – 9)*(1/(2x – 3) – 1/(2x +3).

Solution: (4x 2 – 9)*(1/(2x – 3) – 1/(2x +3) = (2x – 3)(2x + 3)(2x + 3 – 2x + 3)/(2x – 3 )(2x + 3) = 6.

Conclusion

When preparing for school tests, Unified State Examinations and State Examinations, you can always use this material as a hint. Keep in mind that an algebraic expression is a combination of numbers and variables expressed in Latin letters. And also signs of arithmetic operations (addition, subtraction, multiplication, division), parentheses, powers, roots.

Use abbreviated multiplication formulas and knowledge of identities to transform algebraic expressions.

Write us your comments and wishes in the comments - it is important for us to know that you are reading us.

website, when copying material in full or in part, a link to the source is required.

The publication presents the logic of the difference between algebraic expressions for students of basic general and secondary (complete) general education as a transitional stage in the formation of the logic of differences mathematical expressions used in physics, etc. for the further formation of concepts about phenomena, tasks, their classification and the methodology for approaching their solution.

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Algebraic expressions and their characteristics

© Skarzhinsky Y.Kh.

Algebra, as a science, studies the patterns of actions on sets designated by letters.TO algebraic operations include addition, subtraction, multiplication, division, exponentiation, root extraction.As a result of these actions, algebraic expressions were formed.Algebraic expression is an expression consisting of numbers and letters denoting sets with which algebraic operations are performed.These operations were transferred to algebra from arithmetic. In algebra they considerequating one algebraic expression to another, which is their identical equality. Examples of algebraic expressions are given in §1.Methods of transformations and relationships between expressions were also borrowed from arithmetic. Knowledge of arithmetic laws of operations on arithmetic expressions allow you to carry out transformations on similar algebraic expressions, transform them, simplify, compare, analyze.Algebra is the science of patterns of transformations of expressions consisting of sets represented in the form letter designations, interconnected by signs of various actions.There are also more complex algebraic expressions studied in higher education. educational institutions. For now, they can be divided into the types most often used in the school curriculum.

1 Types of algebraic expressions

clause 1 Simple expressions: 4a; (a + b); (a + b)3c; ; .

clause 2 Identical equalities:(a + b)c = ac + bc; ;

item 3 Inequalities: ac ; a + c .

item 4 Formulas: x=2a+5; y=3b; y=0.5d 2 +2;

item 5 Proportions:

First difficulty level

Second difficulty level

Third level of difficultyfrom the point of view of searching for values ​​for sets

a, b, c, m, k, d:

Fourth difficulty levelfrom the point of view of searching for values ​​for sets a, y:

item 6 Equations:

ax+c = -5bx; 4x 2 +2x= 42;

Etc.

clause 7 Functional dependencies: y=3x; y=ax 2 +4b; y=0.5x 2 +2;

Etc.

2 Consider algebraic expressions

2.1 Section 1 presents simple algebraic expressions. There is a view and

more difficult, for example:

As a rule, such expressions do not have the “=” sign. The task when considering such expressions is to transform them and obtain them in a simplified form. When transforming the algebraic expression related to step 1, a new algebraic expression is obtained, which in its meaning is equivalent to the previous one. Such expressions are said to be identically equivalent. Those. the algebraic expression to the left of the equal sign is equivalent in meaning to the algebraic expression to the right. In this case, an algebraic expression of a new type is obtained, called an identical equality (see paragraph 2).

2.2 Section 2 presents algebraic identity equalities, which are formed by algebraic transformation methods, algebraic expressions are considered that are most often used as methods for solving problems in physics. Examples of identical equalities of algebraic transformations, often used in mathematics and physics:

Commutative law of addition: a + b = b + a.

Combination law of addition:(a + b) + c = a + (b + c).

Commutative multiplication law: ab = ba.

Combination law of multiplication:(ab)c = a(bc).

Distributive law of multiplication relative to addition:

(a + b)c = ac + bc.

Distributive law of multiplication relative to subtraction:

(a - b)c = ac - bc.

Identical equalitiesfractional algebraic expressions(assuming that the denominators of the fractions are non-zero):

Identical equalitiesalgebraic expressions with powers:

A) ,

where (n times, ) - integer degree

b) (a + b) 2 =a 2 +2ab+b 2.

Identical equalitiesalgebraic expressions with roots nth degree:

Expression - arithmetic root n th degree from among In particular, - arithmetic square.

Degree with fractional (rational) exponent root:

The equivalent expressions given above are used to transform more complex algebraic expressions that do not contain the “=” sign.

Let's consider an example in which, to transform a more complex algebraic expression, we use knowledge acquired from transforming simpler algebraic expressions in the form of identical equalities.

2.3 Section 3 presents algebraic n equality, in which the algebraic expression of the left side is not equal to the right, i.e. are not identical. In this case, they are inequalities. As a rule, when solving some problems in physics, the properties of inequalities are important:

1) If a, then for any c: a + c .

2) If a and c > 0, then ac .

3) If a and c , then ac > bс .

4) If a , a and b one sign, then 1/a > 1/b .

5) If a and c , then a + c , a - d .

6) If a , c , a > 0, b > 0, c > 0, d > 0, then ac .

7) If a , a > 0, b > 0, then

8) If , then

2.4 Section 4 presents algebraic formulasthose. algebraic expressions in which on the left side of the equal sign there is a letter denoting a set whose value is unknown and must be determined. And on the right side of the equal sign there are sets whose values ​​are known. IN in this case this algebraic expression is called algebraic formula.

An algebraic formula is an algebraic expression containing an equal sign, on the left side of which there is a set whose value is unknown, and on the right side there are sets with known values, based on the conditions of the problem.To determine not known value sets to the left of the “equal” sign, substitute known values ​​of quantities on the right side of the “equal” sign and carry out arithmetic computational operations indicated in the algebraic expression in this part.

Example 1:

Given: Solution:

a=25 Let the algebraic expression be given:

x=? x=2a+5.

This algebraic expression is an algebraic formula because To the left of the equal sign there is a set whose value should be found, and to the right there are sets with known values.

Therefore, it is possible to substitute a known value for the set “a” to determine the unknown value of the set “x”:

x=2·25+5=55. Answer: x=55.

Example 2:

Given: Solution:

a=25 Algebraic expressionis the formula.

b=4 Therefore, it is possible to substitute known

c=8 values ​​for sets to the right of the equal sign,

d=3 to determine the unknown value of the set “k”,

m=20 standing on the left:

n=6 Answer: k=3.2.

QUESTIONS

1 What is an algebraic expression?

2 What types of algebraic expressions do you know?

3 What algebraic expression is called an identity equality?

4 Why is it necessary to know identity equality patterns?

5 What algebraic expression is called a formula?

6 What algebraic expression is called an equation?

7 What algebraic expression is called a functional dependence?

>>Math: Numerical and algebraic expressions

Numeric and algebraic expressions

IN junior classes you learned how to do calculations with whole and fractional numbers, solved equations, got acquainted with geometric shapes, With coordinate plane. All this constituted the content of one school subject "Mathematics". In fact, such an important field of science as mathematics is divided into huge number independent disciplines: algebra, geometry, probability theory, mathematical analysis, mathematical logic, mathematical statistics, game theory, etc. Each discipline has its own objects of study, its own methods of understanding reality.

Algebra, which we are about to study, gives a person the opportunity not only to perform various calculations, but also teaches him to do it as quickly and rationally as possible. Man owning algebraic methods, has an advantage over those who do not master these methods: he calculates faster, navigates more successfully life situations, makes decisions more clearly, thinks better. Our task is to help you master algebraic methods, your task is not to resist learning, to be willing to follow us, overcoming difficulties.

In fact, in elementary school, a window into your life has already been opened. magical world algebra, because algebra primarily studies numerical and algebraic expressions.

Let us recall that a numerical expression is any entry made up of numbers and signs of arithmetic operations (composed, of course, with meaning: for example, 3 + 57 - numeric expression, while 3 + : is not a numeric expression, but a meaningless set of characters). For some reasons (we will talk about them later), letters are often used instead of specific numbers (mainly from Latin alphabet); then an algebraic expression is obtained. These expressions can be very cumbersome. Algebra teaches you to simplify them using different rules, laws, properties, algorithms, formulas, theorems.

Example 1. Simplify a numerical expression:

Solution. Now we will remember something together, and you will see how many algebraic facts you already know. First of all, you need to develop a plan for carrying out the calculations. To do this, you will have to use the conventions accepted in mathematics about the order of operations. Procedure in in this example will be like this:

1) find the value A of the expression in the first brackets:
A = 2.73 + 4.81 + 3.27 - 2.81;

2) find the value B of the expression in the second brackets:

3) divide A by B - then we will know what number C is contained in the numerator (i.e., above the horizontal line);

4) find the value D of the denominator (i.e., the expression contained under the horizontal line):
D = 25 - 37 - 0.4;

5) divide C by D - this will be the desired result. So, there is a calculation plan (and having a plan is half
success!), let's start implementing it.

1) Let's find A = 2.73 + 4.81 + 3.27 - 2.81. Of course, you can count in a row or, as they say, “head to head”: 2.73 + 4.81, then add to this number
3.27, then subtract 2.81. But cultured person It won't calculate that way. He will remember the commutative and associative laws of addition (however, he doesn’t need to remember them, they are always in his head) and will calculate like this:

(2,73 + 3,27) + 4,81 - 2,81) = 6 + 2 = 8.

Now let’s analyze together once again what math facts we had to remember in the process of solving the example (and not just remember, but also use).

1. The order of arithmetic operations.

2. Commutative law of addition: a + b = b + a.

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions