In this lesson we will give a strict definition of a monomial, consider 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 standard actions on monomials, namely reduction to standard view and calculation of specific numerical value 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. 
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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:
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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»:
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So, here is a simplified expression:
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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. 
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3. 
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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 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, 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 , , , .
Monomials are one of the main types of expressions studied in the school algebra course. In this material, we will tell you what these expressions are, define their standard form and show examples, and also understand related concepts, such as the degree of a monomial and its coefficient.
What is a monomial
School textbooks usually give the following definition of this concept:
Definition 1
Monomials include numbers, variables, as well as their powers with natural exponents and different types works compiled from them.
Based on this definition, we can give examples of such expressions. Thus, all numbers 2, 8, 3004, 0, - 4, - 6, 0, 78, 1 4, - 4 3 7 will be monomials. All variables, for example, x, a, b, p, q, t, y, z, will also be monomials by definition. This also includes powers of variables and numbers, for example, 6 3, (− 7, 41) 7, x 2 and t 15, as well as expressions of the form 65 · x, 9 · (− 7) · x · y 3 · 6, x · x · y 3 · x · y 2 · z, etc. Please note that a monomial can contain one number or variable, or several, and they can be mentioned several times in one polynomial.
Such types of numbers as integers, rational numbers, and natural numbers also belong to monomials. You can also include valid and complex numbers. Thus, expressions of the form 2 + 3 · i · x · z 4, 2 · x, 2 · π · x 3 will also be monomials.
What is the standard form of a monomial and how to convert an expression to it
For ease of use, all monomials are first reduced to a special form called standard. Let us formulate specifically what this means.
Definition 2
Standard form of monomial is called its form in which it is the product of a numerical factor and natural degrees different variables. The numerical factor, also called the coefficient of the monomial, is usually written first on the left side.
For clarity, let’s select several monomials of the standard form: 6 (this is a monomial without variables), 4 · a, − 9 · x 2 · y 3, 2 3 5 · x 7. This also includes the expression x y(here the coefficient will be equal to 1), − x 3(here the coefficient is - 1).
Now we give examples of monomials that need to be brought to standard form: 4 a 2 a 3(here you need to combine the same variables), 5 x (− 1) 3 y 2(here you need to combine the numerical factors on the left).
Typically, when a monomial has several variables written in letters, the letter factors are written in alphabetical order. For example, it is preferable to write 6 a b 4 c z 2, how b 4 6 a z 2 c. However, the order may be different if the purpose of the calculation requires it.
Any monomial can be reduced to standard form. To do this, you need to perform all the necessary identity transformations.
The concept of degree of a monomial
The accompanying concept of the degree of a monomial is very important. Let's write down the definition of this concept.
Definition 3
By the power of the monomial, written in standard form, is the sum of the exponents of all variables that are included in its notation. If there are no variables in it, and the monomial itself is different from 0, then its degree will be zero.
Let us give examples of powers of a monomial.
Example 1
Thus, the monomial a has degree equal to 1, since a = a 1. If we have a monomial 7, then it will have degree zero, since it has no variables and is different from 0. And here is the recording 7 a 2 x y 3 a 2 will be a monomial of the 8th degree, because the sum of the exponents of all degrees of the variables included in it will be equal to 8: 2 + 1 + 3 + 2 = 8 .
The monomial reduced to standard form and the original polynomial will have the same degree.
Example 2
We'll show you how to calculate the degree of a monomial 3 x 2 y 3 x (− 2) x 5 y. In standard form it can be written as − 6 x 8 y 4. We calculate the degree: 8 + 4 = 12 . This means that the degree of the original polynomial is also equal to 12.
Concept of monomial coefficient
If we have a monomial reduced to standard form that includes at least one variable, then we talk about it as a product with one numerical factor. This factor is called a numerical coefficient, or monomial coefficient. Let's write down the definition.
Definition 4
The coefficient of a monomial is the numerical factor of a monomial reduced to standard form.
Let's take as an example the coefficients of various monomials.
Example 3
So, in the expression 8 a 3 the coefficient will be the number 8, and in (− 2 , 3) x y z they will − 2 , 3 .
Particular attention should be paid to coefficients equal to one and minus one. As a rule, they are not explicitly indicated. It is believed that in a monomial of the standard form, in which there is no numerical factor, the coefficient is equal to 1, for example, in the expressions a, x · z 3, a · t · x, since they can be considered as 1 · a, x · z 3 – How 1 x z 3 etc.
Similarly, in monomials that do not have a numerical factor and that begin with a minus sign, we can consider - 1 to be the coefficient.
Example 4
For example, the expressions − x, − x 3 · y · z 3 will have such a coefficient, since they can be represented as − x = (− 1) · x, − x 3 · y · z 3 = (− 1) · x 3 y z 3 etc.
If a monomial does not have a single letter factor at all, then we can talk about a coefficient in this case. The coefficients of such monomials-numbers will be these numbers themselves. So, for example, the coefficient of the monomial 9 will be equal to 9.
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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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What does it mean to reduce a monomial to standard form?
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.
How to bring a monomial to 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:
- First, a grouping of numerical factors is performed, as well as identical variables and their powers;
- Secondly, the product of the numbers is calculated and applied.
As a result of applying the stated rule, any monomial will be reduced to a standard form.
Examples, solutions
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 numbers in the first brackets is equal to 6, and the rule for multiplying powers with the same bases 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: .
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.
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
Example 1Given 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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