Find constant numbers in standard form monomial. I

There are many different mathematical expressions in mathematics, and some of them have their own fixed names. We have 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 be included in the product to some extent. 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 singletons. As you can see, a number or a variable alone (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 monomial because they don't fit the definition. The first expression uses "sum", which is not allowed, the second uses "division", and the third uses the difference.

Consider a few more examples.

For example, the expression 2*a^3*b/3 is also a monomial, although division is present there. 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? correctly answer that the first expression is not a monomial, but the second one.

Standard form of a 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 looks 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 monomial coefficient.

In order to bring the 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 the 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, then we get the first monomial. This action is called bringing the 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.

The concept of a monomial

Definition of a monomial: a monomial is algebraic expression, which uses only multiplication.

Standard form of a monomial

What is the standard form of a monomial? The monomial is written in standard form, if it has a numerical factor in the first place and this factor, it 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 occurs 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 occurs only once, they are arranged in the Latin alphabetical order, but where is the coefficient of the monomial, i.e. number factor that should come first? He is here equal to one: 1adm.

Can the monomial coefficient be negative? Yes, maybe, example: -5a.

Can a monomial coefficient be fractional? Yes, maybe, example: 5.2a.

If the monomial consists only of a number, i.e. does not have letters, how to bring it to the standard form? Any monomial that is a number is already in standard form, for example: the number 5 is a standard form monomial.

Reduction of monomials to standard form

How to bring monomial to standard form? Consider examples.

Let the monomial 2a4b be given, we need to bring it to the standard form. We multiply two of its numerical factors and get 8ab. Now the monomial is written in the 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, the product aa is replaced by the second power 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.

An 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? We bring the second monomial to the standard form, we get 10abc. Now it is clear 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 monomials? We can only sum similar monomials. Consider the example of addition of monomials. What is the sum of the monomials 5a and 2a? The sum of these monomials will be a monomial similar to them, the coefficient of which is equal to the sum the coefficients of the terms. So, the sum of the monomials is 5a + 2a = 7a.

More examples of addition of 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 is reduced to adding their coefficients.

Subtraction of monomials

What is the difference of monomials? We can only subtract similar monomials. Consider an example of subtracting monomials. What is the difference between the 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 monomials is equal to 5a - 2a = 3a.

More examples of subtracting monomials:

10a2 - 3a2 = 7a2
5a 2 b 3 c 4 - 3a 2 b 3 c 4 = 2a 2 b 3 c 4

Multiplication of monomials

What is the product of monomials? Consider an example:

those. the product of monomials is equal to the monomial whose factors are composed 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 has “a” in the degree: in the first - “a” in the degree of 2, and in the second - “a” in the degree of 5. This means that the product will have “a” in the degree of 7, because when multiplying identical letters, their exponents add up:

A 2 * a 5 = a 7 .

The same applies to the factor "b".

The coefficient of the first factor is equal to two, and the second - to one, so we get 2 * 1 = 2 as a result.

This is how the result 2a 7 b 12 was calculated.

From these examples, it can be seen that the coefficients of monomials are multiplied, and the same letters are replaced by the sums of their degrees in the product.

Monomials are products of numbers, variables and their powers. Numbers, variables and their degrees 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. The coefficients 1 and -1 are not written, but they retain a minus 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. Monomials are also considered numbers, variables and their degrees.

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 the first place and the powers of various variables. Any monomial can be brought to standard form by multiplying all the variables and numbers included in it. Here is an example of bringing a monomial to the standard form:

4x2y4(-5)yx3 = 4(-5)x2x3y4y = -20x5y5

The numerical factor of a monomial written in standard form is called the coefficient of a monomial. For example, the coefficient of the monomial -7x2y2 is -7. The coefficients of 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 the 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 8x3yz2 monomial is 6, the 6x monomial is 1, and the -10 monomial is 0.

Multiplication of monomials. Raising monomials to a power

When multiplying monomials and raising monomials to a power, the rule of multiplication of powers with the same base and the rule of exponentiation. In this case, a monomial is obtained, which is usually represented in standard form.

for example

4x3y2(-3)x2y = 4(-3)x3x2y2y = -12x5y3

((-5)x3y2)3 = (-5)3x3*3y2*3 = -125x9y6

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 alphabetic 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.

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, 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 natural indicator and different types works made from 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. It can also include real and complex numbers. 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 call it such a form in which it is the product of a numerical factor and natural degrees 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

Very important is related concept monomial degree. 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 the ratios 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 - as 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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