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 view 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.
Degree of a monomial
For a monomial there is the concept of its degree. Let's figure out what it is.
Definition.
Degree of a monomial standard form is the sum of the exponents of all variables included in its record; if there are no variables in the monomial entry, and it is different from zero, then its degree is considered to be zero; the number zero is considered a monomial, the degree of which is not defined.
The definition of the degree of a monomial allows us to give examples. The degree of the monomial a is equal to one, since a is a 1 . The degree of the monomial 5 is zero, since it is non-zero and its notation contains no 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 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 standard form monomial. To illustrate what has been said, we 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, having 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 formalize the above reasoning in the form of a definition.
Definition.
Monomial coefficient is the numerical factor of the monomial written in the standard form.
Now we can give examples of the 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 the coefficient −2.3 .
The coefficients of monomials equal to 1 and −1 deserve special attention. The point here is that they are usually not explicitly present in the record. It is believed that the coefficient of monomials of the standard form, which 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 coefficient 1, since a can be considered as 1 a, x z 3 as 1 x z 3, etc.
Similarly, the coefficient of monomials, whose entries in the standard form do not have a numerical factor and begin with a minus sign, is considered minus one. For example, the monomials −x , −x 3 y z 3, etc. have coefficient −1 , since −x=(−1) x , −x 3 y z 3 =(−1) x 3 y z 3 etc.
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 7 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. 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. Student's textbook educational institutions/ A. G. Mordkovich. - 17th ed., add. - M.: Mnemozina, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
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 grounds. 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 over 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 numeric expression, which 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 , , , .