Definition of a monomial: related concepts, examples. Definition of a monomial, related concepts, examples Standard form of a monomial rule with an example

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.

Monomials are one of the main types of expressions studied as part of a school algebra course. In this article, 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 - 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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Monomial. Definition

Monomial is a mathematical expression that is the product of a prime factor and one or more variables.

Monomials include all numbers, variables, their powers with a natural exponent:
42; 3; 0; 62; 2 3 ; b 3 ; ax4; 4x3; 5a2; 12xyz 3 .

Quite often it is difficult to determine whether a given mathematical expression refers to a monomial or not. For example, $\frac(4a^3)(5)$. Is it monomial or not? To answer this question, we need to simplify the expression, i.e. represent in the form: $\frac(4)(5)*а^3$.
We can say for sure that this expression is a monomial.

Standard form of a monomial

When calculating, it is desirable to bring the monomial to the standard form. This is the shortest and most understandable notation of the monomial.

The order of bringing the monomial to the standard form is as follows:
1. Multiply the coefficients of the monomial (or numerical factors) and put the result in the first place.
2. Select all degrees with the same letter base and multiply them.
3. Repeat point 2 for all variables.

Examples.
I. Reduce the given monomial $3x^2zy^3*5y^2z^4$ to standard form.

Decision.
1. Multiply the coefficients of the monomial $15x^2y^3z * y^2z^4$.
2. Now let us present similar terms $15х^2y^5z^5$.

II. Convert the given monomial $5a^2b^3 * \frac(2)(7)a^3b^2c$ to standard form.

Decision.
1. Multiply the coefficients of the monomial $\frac(10)(7)a^2b^3*a^3b^2c$.
2. Now let us present similar terms $\frac(10)(7)a^5b^5c$.

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:

1. 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.
2. Multiply all identical variables so that each variable occurs only once in the monomial.
3. 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

Decision:

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

Indicator 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 its 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 pm Part 1. A textbook for students of 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.