Monomials are one of the main types of expressions studied within school course algebra. 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
IN school textbooks usually given following definition this concept:
Definition 1
Monomials include numbers, variables, as well as their powers with natural indicator 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 convenience, all monomials first lead to special type, 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).
Usually, in the case 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 the 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 is not a single variable 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 zero degree, because 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 us 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 the 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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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 operations on monomials, namely reduction to a standard form and calculation of a specific numerical value monomial at given values 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 over monomials
Lesson:The concept of a monomial. Standard view monomial
Consider some examples:
3. 
;
We'll 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 definition of monomial : a monomial is called something like this algebraic expression, which 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 for 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 monomial coefficient 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, then we have arithmetic numeric expression, which should be calculated. 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 given example it is required to calculate the value of the monomial at , , , .
Power of a monomial
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.
Monomial coefficient
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.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Mordkovich A. G. Algebra. 7th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
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 appears 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 appears 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 monomials 5a and 2a? The sum of these monomials will be a monomial similar to them, whose coefficient equal to the sum 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.
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 a 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 powers, the rule of multiplying powers is used with the same basis and the rule for raising a degree to a degree. 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