Monomials are one of the main types of expressions studied as part of a school algebra course. In this material, we will tell you what these expressions are, define their standard form and show examples, as well as deal with 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
Monomers include numbers, variables, as well as their degrees with a natural indicator, and different types of products made up of them.
Based on this definition, we can give examples of such expressions. So, all numbers 2 , 8 , 3004 , 0 , - 4 , - 6 , 0 , 78 , 1 4 , - 4 3 7 will refer to monomials. All variables, for example, x , a , b , p , q , t , y , z will also be monomials by definition. This also includes the powers of variables and numbers, for example, 6 3 , (− 7 , 41) 7 , x 2 and t 15, as well as expressions like 65 x , 9 (− 7) x y 3 6 , x x y 3 x y 2 z etc. Please note that a monomial can include either one number or variable, or several, and they can be mentioned several times as part of one polynomial.
Such types of numbers as integers, rationals, naturals also belong to monomials. You can also include real and complex numbers here. So, expressions like 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 of work, all monomials are first reduced to a special form, called the standard one. Let's be specific about what this means.
Definition 2
The standard form of the monomial they call it such a form in which it is the product of a numerical factor and natural powers of different variables. The numerical factor, also called the monomial coefficient, is usually written first from the left side.
For clarity, we 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 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, the preferred entry 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 computation requires it.
Any monomial can be reduced to standard form. To do this, you need to perform all the necessary identical transformations.
The concept of the degree of a monomial
The accompanying notion of the degree of a monomial is very important. Let us write down the definition of this concept.
Definition 3
Degree of a monomial, written in standard form, is the sum of the exponents of all variables that are included in its record. 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 the degrees of the monomial.
Example 1
So, monomial a has degree 1 because a = a 1 . If we have a monomial 7 , then it will have a zero degree, since it has no variables and is different from 0 . And here is the entry 7 a 2 x y 3 a 2 will be a monomial of the 8th degree, because the sum of the exponents of all the degrees of the variables included in it will be equal to 8: 2 + 1 + 3 + 2 = 8 .
The standardized monomial and the original polynomial will have the same degree.
Example 2
Let's show 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 . Hence, the degree of the original polynomial is also equal to 12 .
The concept of a monomial coefficient
If we have a standardized monomial that includes at least one variable, then we talk about it as a product with one numerical factor. This factor is called the numerical coefficient, or the 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.
Take, for 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 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 the coefficient - 1.
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 literal multiplier at all, then it is possible to talk about a coefficient in this case as well. 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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1. An integer positive coefficient. Let we have the monomial +5a, since the positive number +5 is considered to be the same as the arithmetic number 5, then
5a = a ∙ 5 = a + a + a + a + a.
Also +7xy² = xy² ∙ 7 = xy² + xy² + xy² + xy² + xy² + xy² + xy²; +3a³ = a³ ∙ 3 = a³ + a³ + a³; +2abc = abc ∙ 2 = abc + abc and so on.
Based on these examples, we can establish that a positive integer coefficient shows how many times the literal factor (or: the product of literal factors) of the monomial is repeated by the term.
One should get used to this to such an extent that it immediately appears in the imagination that, for example, in the polynomial
3a + 4a² + 5a³
the matter is reduced to the fact that first a² is repeated 3 times as a term, then a³ is repeated 4 times as a term, and then a is repeated 5 times as a term.
Also: 2a + 3b + c = a + a + b + b + b + c
x³ + 2xy² + 3y³ = x³ + xy² + xy² + y³ + y³ + y³ etc.
2. Positive fractional coefficient. Let we have the monomial +a. Since the positive number + coincides with the arithmetic number, then +a = a ∙ , which means: you need to take three fourths of the number a, i.e.
Therefore: a fractional positive coefficient shows how many times and what part of the literal multiplier of the monomial is repeated by the term.
Polynomial 
should be easily represented as:
etc.
3. Negative coefficient. Knowing the multiplication of relative numbers, we can easily establish that, for example, (+5) ∙ (–3) = (–5) ∙ (+3) or (–5) ∙ (–3) = (+5) ∙ (+ 3) or in general a ∙ (–3) = (–a) ∙ (+3); also a ∙ (–) = (–a) ∙ (+), etc.
Therefore, if we take a monomial with a negative coefficient, for example, –3a, then
–3a = a ∙ (–3) = (–a) ∙ (+3) = (–a) ∙ 3 = – a – a – a (–a is taken as a term 3 times).
From these examples, we see that the negative coefficient shows how many times the letter part of the monomial, or its certain fraction, taken with a minus sign, is repeated by the term.
In this lesson, we will give a strict definition of a monomial, consider various examples from the textbook. Recall the rules for multiplying powers with the same base. Let us give a definition of the standard form of a monomial, the coefficient of a monomial, and its literal part. Let's consider two basic 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 the rule for reducing the monomial to the standard form. Let's learn how to solve typical problems with any monomials.
Topic:monomials. Arithmetic operations on monomials
Lesson:The concept of a monomial. Standard form of a monomial
Consider some examples:
3. 
;
Let's 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 a monomial : a monomial is an algebraic expression that consists of a 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 operations of addition, subtraction or division, while in examples 1-3, which are monomials, these operations are not.
Here are a few more examples:
Expression number 8 is a monomial, since it is the product of a power and a number, while example 9 is not a monomial.
Now let's find out actions on monomials .
1. Simplification. Consider example #3 ;and example #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 instance, as “”, similarly, the variables “” and “” occur 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 "" occurs twice. That is, this expression should be simplified, thus, we come to the first action performed on monomials is to bring the monomial to the standard form . To do this, we bring the expression from Example 3 to the standard form, then we define this operation and learn how to bring any monomial to the standard form.
So consider an example:
The first step in the standardization operation is always to multiply all numeric factors:
;
The result of this action will be called monomial coefficient .
Next, you need to multiply the degrees. We multiply the degrees of the variable " X”according to the rule for multiplying powers with the same base, which states that when multiplied, the exponents add up:
Now let's multiply the powers at»:
;
So here's a simplified expression:
;
Any monomial can be reduced to standard form. Let's formulate standardization rule :
Multiply all numerical factors;
Put 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 having the same letter part are called similar.
Now you need to earn technique for reducing monomials to standard form . Consider examples from the textbook:
Task: bring the monomial to the standard form, name the coefficient and the letter part.
To complete the task, we use the rule of bringing the monomial to the standard form and the properties of the degrees.
1. 
;
3. 
;
Comments on the first example: To begin with, let's determine whether this expression is really a monomial, for this we check if it contains multiplication operations of numbers and powers and whether it contains addition, subtraction or division operations. We can say that this expression is a monomial, since the above condition is satisfied. Further, according to the rule of bringing the monomial to the standard form, we multiply the numerical factors:
- we have found the coefficient of the given monomial;
; ; ; that is, the literal part of the expression is received:;
write down the answer: ;
Comments on the second example: Following the rule, we execute:
1) multiply numerical factors:
2) multiply the powers:
Variables and are presented in a single copy, that is, they cannot be multiplied with anything, they are rewritten without changes, the degree is multiplied:
write down the answer:
;
In this example, the monomial coefficient is equal to one, and the literal part is .
Comments on the third example: a similarly to the previous examples, we perform the following actions:
1) multiply numerical factors:
;
2) multiply the powers:
;
write out the answer: ;
In this case, the coefficient of the monomial is equal to "", and the literal part 
.
Now consider second standard operation on monomials . Since a monomial is an algebraic expression consisting of literal variables that can take on specific numerical values, we have an arithmetic numerical expression that should be calculated. That is, the following operation on polynomials is calculating their specific numerical value .
Consider an example. The monomial is given:
this monomial has already been reduced to standard form, its coefficient is equal to one, and the literal part
Earlier we said that an algebraic expression cannot always be calculated, that is, the variables that enter it may not take 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, it is required to calculate the value of the monomial for , , , .
Monomial is an expression that is the product of two or more factors, each of which is a number expressed by a letter, digits, or power (with a non-negative integer exponent):
2a, a 3 x, 4abc, -7x
Since the product of identical factors can be written as a degree, then a single degree (with a non-negative integer exponent) is also a monomial:
(-4) 3 , x 5 ,
Since a number (whole or fractional), expressed by a letter or numbers, can be written as the product of this number by one, then any single number can also be considered as a monomial:
x, 16, -a,
Standard form of a monomial
Standard form of a monomial- this is a monomial, which has only one numerical factor, which must be written in the first place. All variables are in alphabetical order and are contained in the monomial only once.
Numbers, variables, and degrees of variables also refer to monomials of the standard form:
7, b, x 3 , -5b 3 z 2 - monomials of standard form.
The numerical factor of a standard form monomial is called monomial coefficient. Monomial coefficients equal to 1 and -1 are usually not written.
If there is no numerical factor in the monomial of the standard form, then it is assumed that the coefficient of the monomial is 1:
x 3 = 1 x 3
If there is no numerical factor in the monomial of the standard form and it is preceded by a minus sign, then it is assumed that the coefficient of the monomial is -1:
-x 3 = -1 x 3
Reduction of a monomial to standard form
To bring the monomial to standard form, you need:
- Multiply numerical factors, if there are several. Raise a numeric factor to a power if it has an exponent. Put the number multiplier in first place.
- Multiply all identical variables so that each variable occurs only once in the monomial.
- Arrange variables after the numeric factor in alphabetical order.
Example. Express the monomial in standard form:
a) 3 yx 2 (-2) y 5 x; b) 6 bc 0.5 ab 3
Solution:
a) 3 yx 2 (-2) y 5 x= 3 (-2) x 2 xyy 5 = -6x 3 y 6
b) 6 bc 0.5 ab 3 = 6 0.5 abb 3 c = 3ab 4 c
Degree of a monomial
Degree of a monomial is the sum of the exponents of all the letters in it.
If a monomial is a number, that is, it does not contain variables, then its degree is considered equal to zero. For example:
5, -7, 21 - zero degree monomials.
Therefore, to find the degree of a monomial, you need to determine the exponent of each of the letters included in it and add these exponents. If the exponent of the letter is not specified, then it is equal to one.
Examples:
So how are u x the exponent is not specified, which means it is equal to 1. The monomial does not contain other variables, which means that its degree is equal to 1.
The monomial contains only one variable in the second degree, which means that the degree of this monomial is 2.
3) ab 3 c 2 d
Index a is equal to 1, the indicator b- 3, indicator c- 2, indicator d- 1. The degree of this monomial is equal to the sum of these indicators.
