For a monomial there is the concept of its degree. Let's figure out what it is.
Definition.
Power of a monomial standard form is the sum of exponents of all variables included in its record; if there are no variables in the notation of a monomial and it is different from zero, then its degree is considered equal to zero; the number zero is considered a monomial whose degree is undefined.
Determining the degree of a monomial allows you to give examples. The degree of the monomial a is equal to one, since a is a 1. The power of the monomial 5 is zero, since it is non-zero and its notation does not contain variables. And the product 7·a 2 ·x·y 3 ·a 2 is a monomial of the eighth degree, since the sum of the exponents of all variables a, x and y is equal to 2+1+3+2=8.
By the way, the degree of a monomial not written in standard form, is equal to the degree of the corresponding monomial of standard form. To illustrate this, let us calculate the degree of the monomial 3 x 2 y 3 x (−2) x 5 y. This monomial in standard form has the form −6·x 8 ·y 4, its degree is 8+4=12. Thus, the degree of the original monomial is 12.
A monomial in standard form, which has at least one variable in its notation, is a product with a single numerical factor - a numerical coefficient. This coefficient is called the monomial coefficient. Let us formulate the above arguments in the form of a definition.
Definition.
Monomial coefficient is the numerical factor of a monomial written in standard form.
Now we can give examples of coefficients of various monomials. The number 5 is the coefficient of the monomial 5·a 3 by definition, similarly the monomial (−2,3)·x·y·z has a coefficient of −2,3.
The coefficients of the monomials, equal to 1 and −1, deserve special attention. The point here is that they are usually not explicitly present in the recording. It is believed that the coefficient of standard form monomials that do not have a numerical factor in their notation is equal to one. For example, monomials a, x·z 3, a·t·x, etc. have a coefficient of 1, since a can be considered as 1·a, x·z 3 - as 1·x·z 3, etc.
Similarly, the coefficient of monomials, the entries of which in standard form do not have a numerical factor and begin with a minus sign, is considered to be minus one. For example, monomials −x, −x 3 y z 3, etc. have a coefficient −1, since −x=(−1) x, −x 3 y z 3 =(−1) x 3 y z 3 and so on.
By the way, the concept of the coefficient of a monomial is often referred to as monomials of the standard form, which are numbers without letter factors. The coefficients of such monomials-numbers are considered to be these numbers. So, for example, the coefficient of the monomial 7 is considered equal to 7.
Bibliography.
Monomials are products of numbers, variables and their powers. Numbers, variables and their powers are also considered monomials. For example: 12ac, -33, a^2b, a, c^9. The monomial 5aa2b2b can be reduced to the form 20a^2b^2. This form is called the standard form of the monomial. That is, the standard form of the monomial is the product of the coefficient (which comes first) and the powers of the variables. Coefficients 1 and -1 are not written, but a minus is kept from -1. Monomial and its standard form
The expressions 5a2x, 2a3(-3)x2, b2x are products of numbers, variables and their powers. Such expressions are called monomials. Numbers, variables and their powers are also considered monomials.
For example, the expressions 8, 35,y and y2 are monomials.
The standard form of a monomial is a monomial in the form of the product of a numerical factor in first place and powers of various variables. Any monomial can be reduced to a standard form by multiplying all the variables and numbers included in it. Here is an example of reducing a monomial to standard form:
4x2y4(-5)yx3 = 4(-5)x2x3y4y = -20x5y5
The numerical factor of a monomial written in standard form is called the coefficient of the monomial. For example, the coefficient of the monomial -7x2y2 is equal to -7. The coefficients of the monomials x3 and -xy are considered equal to 1 and -1, since x3 = 1x3 and -xy = -1xy
The degree of a monomial is the sum of the exponents of all the variables included in it. If a monomial does not contain variables, that is, it is a number, then its degree is considered equal to zero.
For example, the degree of the monomial 8x3yz2 is 6, the monomial 6x is 1, and the degree of -10 is 0.
Multiplying monomials. Raising monomials to powers
When multiplying monomials and raising monomials to a power, the rule for multiplying powers with the same base and the rule for raising a power to a power are used. This produces a monomial, which is usually represented in standard form.
For example
4x3y2(-3)x2y = 4(-3)x3x2y2y = -12x5y3
((-5)x3y2)3 = (-5)3x3*3y2*3 = -125x9y6
We noted that any monomial can be bring to standard form. In this article we will understand what is called bringing a monomial to standard form, what actions allow this process to be carried out, and consider solutions to examples with detailed explanations.
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It is convenient to work with monomials when they are written in standard form. However, quite often monomials are specified in a form different from the standard one. In these cases, you can always go from the original monomial to a monomial of the standard form by performing identity transformations. The process of carrying out such transformations is called reducing a monomial to a standard form.
Let us summarize the above arguments. Reduce the monomial to standard form- this means performing identical transformations with it so that it takes on a standard form.
It's time to figure out how to reduce monomials to standard form.
As is known from the definition, monomials of non-standard form are products of numbers, variables and their powers, and possibly repeating ones. And a monomial of the standard form can contain in its notation only one number and non-repeating variables or their powers. Now it remains to understand how to bring products of the first type to the type of the second?
To do this you need to use the following the rule for reducing a monomial to standard form consisting of two steps:
As a result of applying the stated rule, any monomial will be reduced to a standard form.
All that remains is to learn how to apply the rule from the previous paragraph when solving examples.
Example.
Reduce the monomial 3 x 2 x 2 to standard form.
Solution.
Let's group numerical factors and factors with a variable x. After grouping, the original monomial will take the form (3·2)·(x·x 2) . The product of the numbers in the first brackets is equal to 6, and the rule for multiplying powers with on the same grounds allows the expression in the second brackets to be represented as x 1 +2=x 3. As a result, we obtain a polynomial of the standard form 6 x 3.
Here is a short summary of the solution: 3 x 2 x 2 =(3 2) (x x 2)=6 x 3.
Answer:
3 x 2 x 2 =6 x 3.
So, to bring a monomial to a standard form, you need to be able to group factors, multiply numbers, and work with powers.
To consolidate the material, let's solve one more example.
Example.
Present the monomial in standard form and indicate its coefficient.
Solution.
The original monomial has a single numerical factor in its notation −1, let's move it to the beginning. After this, we will separately group the factors with the variable a, separately with the variable b, and there is nothing to group the variable m with, we will leave it as is, we have 
. After performing operations with powers in brackets, the monomial will take the standard form we need, from which we can see the coefficient of the monomial equal to −1. Minus one can be replaced with a minus sign: .
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 to solve typical tasks 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 appears 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:
;
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 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 in this case the coefficient of the monomial is "", and the literal 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, next operation over polynomials consists of 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.
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 there is a need to implement identity transformations to bring a given monomial into 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.
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:
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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