The concept of a monomial and its standard form. Lesson "The concept of a monomial

In this lesson we will give a strict definition of a monomial and look at various examples from the textbook. Let us recall the rules for multiplying powers with the same bases. Let us define the standard form of a monomial, the coefficient of the monomial and its letter part. Let's consider two main typical operations on monomials, namely reduction to a standard form and calculation of a specific numerical value of a monomial for given values ​​of the literal variables included in it. Let us formulate a rule for reducing a monomial to standard form. Let's learn how to solve standard problems with any monomials.

Subject:Monomials. Arithmetic operations on monomials

Lesson:The concept of a monomial. Standard form of monomial

Consider some examples:

3. ;

Let us find common features for the given expressions. In all three cases, the expression is the product of numbers and variables raised to a power. Based on this we give monomial definition : A monomial is an algebraic expression that consists of the product of powers and numbers.

Now we give examples of expressions that are not monomials:

Let us find the difference between these expressions and the previous ones. It consists in the fact that in examples 4-7 there are addition, subtraction or division operations, while in examples 1-3, which are monomials, there are no these operations.

Here are a few more examples:

Expression number 8 is a monomial because it is the product of a power and a number, whereas example 9 is not a monomial.

Now let's find out actions on monomials .

1. Simplification. Let's look at example No. 3 ;and example No. 2 /

In the second example we see only one coefficient - , each variable occurs only once, that is, the variable " A" is represented in a single copy as "", similarly, the variables "" and "" appear only once.

In example No. 3, on the contrary, there are two different coefficients - and , we see the variable "" twice - as "" and as "", similarly, the variable "" appears twice. That is, this expression should be simplified, thus we arrive at the first action performed on monomials is to reduce the monomial to standard form . To do this, we will reduce the expression from Example 3 to standard form, then we will define this operation and learn how to reduce any monomial to standard form.

So, consider an example:

The first action in the operation of reduction to standard form is always to multiply all numerical factors:

;

The result of this action will be called coefficient of the monomial .

Next you need to multiply the powers. Let's multiply the powers of the variable " X"according to the rule for multiplying powers with the same bases, which states that when multiplying, the exponents are added:

Now let's multiply the powers " at»:

;

So, here is a simplified expression:

;

Any monomial can be reduced to standard form. Let's formulate standardization rule :

Multiply all numerical factors;

Place the resulting coefficient in first place;

Multiply all degrees, that is, get the letter part;

That is, any monomial is characterized by a coefficient and a letter part. Looking ahead, we note that monomials that have the same letter part are called similar.

Now we need to work out technique for reducing monomials to standard form . Consider examples from the textbook:

Assignment: bring the monomial to standard form, name the coefficient and the letter part.

To complete the task, we will use the rule for reducing a monomial to a standard form and the properties of powers.

1. ;

3. ;

Comments on the first example: First, let's determine whether this expression is really a monomial; to do this, let's check whether it contains operations of multiplication of numbers and powers and whether it contains operations of addition, subtraction or division. We can say that this expression is a monomial since the above condition is satisfied. Next, according to the rule for reducing a monomial to a standard form, we multiply the numerical factors:

- we found the coefficient of a given monomial;

; ; ; that is, the literal part of the expression is obtained:;

Let's write down the answer: ;

Comments on the second example: Following the rule we perform:

1) multiply numerical factors:

2) multiply the powers:

Variables are presented in a single copy, that is, they cannot be multiplied with anything, they are rewritten without changes, the degree is multiplied:

Let's write down the answer:

;

In this example, the coefficient of the monomial is equal to one, and the letter part is .

Comments on the third example: a Similar to the previous examples, we perform the following actions:

1) multiply numerical factors:

;

2) multiply the powers:

;

Let's write down the answer: ;

In this case, the coefficient of the monomial is “”, and the letter part .

Now let's consider second standard operation on monomials . Since a monomial is an algebraic expression consisting of literal variables that can take on specific numeric values, we have an arithmetic numeric expression that must be evaluated. That is, the next operation on polynomials is calculating their specific numerical value .

Let's look at an example. Monomial given:

this monomial has already been reduced to standard form, its coefficient is equal to one, and the letter part

Earlier we said that an algebraic expression cannot always be calculated, that is, the variables that are included in it cannot take on any value. In the case of a monomial, the variables included in it can be any; this is a feature of the monomial.

So, in the given example, you need to calculate the value of the monomial at , , , .

Basic information about monomials contains the clarification that any monomial can be reduced to a standard form. In the material below we will look at this issue in more detail: we will outline the meaning of this action, define the steps that allow us to set the standard form of a monomial, and also consolidate the theory by solving examples.

The meaning of reducing a monomial to standard form

Writing a monomial in standard form makes it more convenient to work with it. Often monomials are specified in a non-standard form, and then it becomes necessary to carry out identical transformations to bring the given monomial into a standard form.

Definition 1

Reducing a monomial to standard form is the performance of appropriate actions (identical transformations) with a monomial in order to write it in standard form.

Method for reducing a monomial to standard form

From the definition it follows that a monomial of a non-standard form is a product of numbers, variables and their powers, and their repetition is possible. In turn, a monomial of the standard type contains in its notation only one number and non-repeating variables or their powers.

To bring a non-standard monomial into standard form, you must use the following rule for reducing a monomial to standard form:

• the first step is to group numerical factors, identical variables and their powers;
• the second step is to calculate the products of numbers and apply the property of powers with equal bases.

Examples and their solutions

Given a monomial 3 x 2 x 2 . It is necessary to bring it to a standard form.

Solution

Let us group numerical factors and factors with variable x, as a result the given monomial will take the form: (3 2) (x x 2) .

The product in parentheses is 6. Applying the rule of multiplication of powers with the same bases, we present the expression in brackets as: x 1 + 2 = x 3. As a result, we obtain a monomial of the standard form: 6 x 3.

A short version of the solution looks like this: 3 · x · 2 · x 2 = (3 · 2) · (x · x 2) = 6 · x 3 .

Answer: 3 x 2 x 2 = 6 x 3.

Example 2

The monomial is given: a 5 · b 2 · a · m · (- 1) · a 2 · b . It is necessary to bring it into a standard form and indicate its coefficient.

Solution

the given monomial has one numerical factor in its notation: - 1, let’s move it to the beginning. Then we will group the factors with the variable a and the factors with the variable b. There is nothing to group the variable m with, so we leave it in its original form. As a result of the above actions we get: - 1 · a 5 · a · a 2 · b 2 · b · m.

Let's perform operations with powers in brackets, then the monomial will take the standard form: (- 1) · a 5 + 1 + 2 · b 2 + 1 · m = (- 1) · a 8 · b 3 · m. From this entry we can easily determine the coefficient of the monomial: it is equal to - 1. It is quite possible to replace minus one simply with a minus sign: (- 1) · a 8 · b 3 · m = - a 8 · b 3 · m.

A short record of all actions looks like this:

a 5 b 2 a m (- 1) a 2 b = (- 1) (a 5 a a 2) (b 2 b) m = = (- 1) a 5 + 1 + 2 b 2 + 1 m = (- 1) a 8 b 3 m = - a 8 b 3 m

Answer:

a 5 · b 2 · a · m · (- 1) · a 2 · b = - a 8 · b 3 · m, the coefficient of the given monomial is - 1.

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I. Expressions that are made up of numbers, variables and their powers using the action of multiplication are called monomials.

Examples of monomials:

A) a; b) ab; V) 12; G)-3c; e) 2a 2 ∙(-3.5b) 3 ; e)-123.45xy 5 z; and) 8ac∙2.5a 2 ∙(-3c 3).

II. This type of monomial, when the numerical factor (coefficient) comes first, followed by the variables with their powers, is called the standard type of monomial.

Thus, the monomials given above, under the letters a B C), G) And e) written in standard form, and the monomials under the letters e) And and) it is required to bring it to a standard form, i.e. to a form where the numerical factor comes first, followed by the letter factors with their exponents, and the letter factors are in alphabetical order. Let us present monomials e) And and) to the standard view.

e) 2a 2 ∙(-3.5b) 3=2a 2 ∙(-3.5) 3 ∙b 3 =-2a 2 ∙3.5∙3.5∙3.5∙b 3 = -85.75a 2 b 3 ;

and) 8ac∙2.5a 2 ∙(-3c 3)=-8∙2.5∙3a 3 c 3 = -60a 3 c 3 .

III.The sum of the exponents of all variables included in a monomial is called the degree of the monomial.

Examples. What degree do monomials have? a) - g)?

a) a. First;

b) ab. Second: A in the first degree and b to the first power - the sum of indicators 1+1=2 ;

V) 12. Zero, since there are no letter factors;

G) -3c. First;

e) -85.75a 2 b 3 . Fifth. We have reduced this monomial to standard form, we have A to the second degree and b in the third. Let's add up the indicators: 2+3=5 ;

e) -123.45xy 5 z. Seventh. We added up the exponents of the letter factors: 1+5+1=7 ;

and) -60a 3 c 3 . Sixth, since the sum of the exponents of the letter factors 3+3=6 .

IV. Monomials that have the same letter part are called similar monomials.

Example. Indicate similar monomials among the given monomials 1) -7).

1) 3aabbc; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 2 bac; 5) 10aaa 2 x; 6) -2.3a 4 x; 7) 34x 2 y.

Let us present monomials 1), 4) And 5) to the standard view. Then the line of monomials data will look like this:

1) 3a 2 b 2 c; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 3 bc; 5) 10a 4x; 6) -2.3a 4 x; 7) 34x 2 y.

Similar will be those that have the same letter part, i.e. 1) and 3) ; 2) and 4); 5) and 6).

1) 3a 2 b 2 c and 3) 56a 2 b 2 c;

2) -4.1a 3 bc and 4) 98.7a 3 bc;

5) 10a 4 x and 6) -2.3a 4 x.

The concept of a monomial

Definition of a monomial: A monomial is an algebraic expression that uses only multiplication.

Standard form of monomial

What is the standard form of a monomial? A monomial is written in standard form, if it has a numerical factor in the first place and this factor is called the coefficient of the monomial, there is only one in the monomial, the letters of the monomial are arranged in alphabetical order and each letter appears only once.

An example of a monomial in standard form:

here in the first place is the number, the coefficient of the monomial, and this number is only one in our monomial, each letter occurs only once and the letters are arranged in alphabetical order, in this case it is the Latin alphabet.

Another example of a monomial in standard form:

each letter occurs only once, they are arranged in Latin alphabetical order, but where is the coefficient of the monomial, i.e. the numeric factor that should come first? Here it is equal to one: 1adm.

Can the coefficient of a monomial be negative? Yes, maybe, example: -5a.

Can the coefficient of a monomial be fractional? Yes, maybe, example: 5.2a.

If a monomial consists only of a number, i.e. has no letters, how can I bring it to standard form? Any monomial that is a number is already in standard form, for example: the number 5 is a monomial in standard form.

Reducing monomials to standard form

How to bring a monomial to standard form? Let's look at examples.

Let the monomial 2a4b be given; we need to bring it to standard form. We multiply its two numerical factors and get 8ab. Now the monomial is written in standard form, i.e. has only one numerical factor, written in the first place, each letter in the monomial occurs only once and these letters are arranged in alphabetical order. So 2a4b = 8ab.

Given: monomial 2a4a, bring the monomial to standard form. We multiply the numbers 2 and 4, replacing the product aa with the second power of a 2. We get: 8a 2 . This is the standard form of this monomial. So 2a4a = 8a 2 .

Similar monomials

What are similar monomials? If monomials differ only in coefficients or are equal, then they are called similar.

Example of similar monomials: 5a and 2a. These monomials differ only in coefficients, which means they are similar.

Are the monomials 5abc and 10cba similar? Let's bring the second monomial to standard form and get 10abc. Now we can see that the monomials 5abc and 10abc differ only in their coefficients, which means that they are similar.

Addition of monomials

What is the sum of the monomials? We can only sum similar monomials. Let's look at an example of adding monomials. What is the sum of the monomials 5a and 2a? The sum of these monomials will be a monomial similar to them, the coefficient of which is equal to the sum of the coefficients of the terms. So, the sum of the monomials is 5a + 2a = 7a.

More examples of adding monomials:

2a 2 + 3a 2 = 5a 2
2a 2 b 3 c 4 + 3a 2 b 3 c 4 = 5a 2 b 3 c 4

Again. You can only add similar monomials; addition comes down to adding their coefficients.

Subtracting monomials

What is the difference between the monomials? We can only subtract similar monomials. Let's look at an example of subtracting monomials. What is the difference between monomials 5a and 2a? The difference of these monomials will be a monomial similar to them, the coefficient of which is equal to the difference of the coefficients of these monomials. So, the difference of the monomials is 5a - 2a = 3a.

More examples of subtracting monomials:

10a 2 - 3a 2 = 7a 2
5a 2 b 3 c 4 - 3a 2 b 3 c 4 = 2a 2 b 3 c 4

Multiplying monomials

What is the product of monomials? Let's look at an example:

those. the product of monomials is equal to a monomial whose factors are made up of the factors of the original monomials.

Another example:

2a 2 b 3 * a 5 b 9 = 2a 7 b 12 .

How did this result come about? Each factor contains “a” to the power: in the first - “a” to the power of 2, and in the second - “a” to the power of 5. This means that the product will contain “a” to the power of 7, because when multiplying identical letters, the exponents of their powers fold up:

A 2 * a 5 = a 7 .

The same applies to the factor “b”.

The coefficient of the first factor is two, and the second is one, so the result is 2 * 1 = 2.

This is how the result was calculated: 2a 7 b 12.

From these examples it is clear that the coefficients of monomials are multiplied, and identical letters are replaced by the sums of their powers in the product.

There are many different mathematical expressions in mathematics, and some of them have their own names. We are about to get acquainted with one of these concepts - this is a monomial.

A monomial is a mathematical expression that consists of a product of numbers, variables, each of which can appear in the product to some degree. In order to better understand the new concept, you need to familiarize yourself with several examples.

Examples of monomials

Expressions 4, x^2 , -3*a^4, 0.7*c, ¾*y^2 are monomials. As you can see, just one number or variable (with or without a power) is also a monomial. But, for example, the expressions 2+с, 3*(y^2)/x, a^2 –x^2 are already are not monomials, since they do not fit the definitions. The first expression uses “sum,” which is unacceptable, the second uses “division,” and the third uses difference.

Let's consider a few more examples.

For example, the expression 2*a^3*b/3 is also a monomial, although there is division involved. But in this case, division occurs by a number, and therefore the corresponding expression can be rewritten as follows: 2/3*a^3*b. One more example: Which of the expressions 2/x and x/2 is a monomial and which is not? The correct answer is that the first expression is not a monomial, but the second is a monomial.

Standard form of monomial

Look at the following two monomial expressions: ¾*a^2*b^3 and 3*a*1/4*b^3*a. In fact, these are two identical monomials. Isn't it true that the first expression seems more convenient than the second?

The reason for this is that the first expression is written in standard form. The standard form of a polynomial is a product made up of a numerical factor and powers of various variables. The numerical factor is called the coefficient of the monomial.

In order to bring a monomial to its standard form, it is enough to multiply all the numerical factors present in the monomial and put the resulting number in first place. Then multiply all powers that have the same letter base.

Reducing a monomial to its standard form

If in our example in the second expression we multiply all the numerical factors 3*1/4 and then multiply a*a, we get the first monomial. This action is called reducing a monomial to its standard form.

If two monomials differ only by a numerical coefficient or are equal to each other, then such monomials are called similar in mathematics.