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 power, a single power (with a non-negative integer exponent) is also a monomial:
(-4) 3 , x 5 ,
Since a number (integer or fraction), expressed by a letter or numbers, can be written as the product of this number by one, any individual number can also be considered as a monomial:
x, 16, -a,
Standard form of monomial
Standard form of monomial is a monomial that has only one numerical factor, which must be written in first place. All variables are in alphabetical order and are contained in a monomial only once.
Numbers, variables and powers of variables also belong to monomials of the standard form:
7, b, x 3 , -5b 3 z 2 - monomials of standard form.
The numerical factor of a monomial of standard form is called coefficient of the monomial. Monomial coefficients equal to 1 and -1 are usually not written.
If a monomial of standard form does not have a numerical factor, then it is assumed that the coefficient of the monomial is equal to 1:
x 3 = 1 x 3
If a monomial of standard form does not have a numerical factor and is preceded by a minus sign, then it is assumed that the coefficient of the monomial is equal to -1:
-x 3 = -1 · x 3
Reducing a monomial to standard form
To bring a monomial to standard form you need to:
- Multiply numerical factors if there are several of them. Raise a numeric factor to a power if it has an exponent. Put the numeric factor first.
- Multiply all the same variables so that each variable appears only once in the monomial.
- Arrange the variables after the numeric factor in alphabetical order.
Example. Present 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
Power of a monomial
Power of a monomial is the sum of the exponents of all letters included 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 are monomials of zero degree.
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 a 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 its degree is equal to 1.
A monomial contains only one variable to the second power, which means the degree of this monomial is 2.
3) ab 3 c 2 d
Index a equals 1, exponent b- 3, indicator c- 2, indicator d- 1. The degree of this monomial is equal to the sum of these indicators.
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.
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: .
The concept of a monomial
Definition of a monomial: a monomial is algebraic expression, which only uses 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 appears 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 occurs 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 occurs 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.
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 there is a need to implement identity transformations to bring a given monomial into 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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