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 doing the following with him identity transformations so that he accepts standard view.
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 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: .
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.
Monomial is mathematical expression, which consists of a product of numbers, variables, each of which can be included 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 in this case division occurs by a number, and therefore the corresponding expression can be rewritten in the following way: 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.
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 to solve typical tasks with any monomials.
Subject:Monomials. Arithmetic operations over monomials
Lesson:The concept of a monomial. Standard form of 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 monomial definition : 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 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 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 this case, the coefficient of the monomial is “”, and the letter 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 numerical values, we have the 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.
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 how to solve standard problems 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 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 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 this case, the coefficient of the monomial is “”, and the letter 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 , , , .