Definition of a monomial, related concepts, examples. The concept of a monomial. Standard form of a monomial What is the square of a monomial

1. Positive integer coefficient. Let us have a monomial +5a, since the positive number +5 is considered to coincide with 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 the positive integer coefficient shows how many times the letter factor (or: product of letter factors) of a monomial is repeated by the addend.

You should get used to this to such an extent that you immediately imagine in your imagination that, for example, in a polynomial

3a + 4a² + 5a³

the matter boils down 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 us have a monomial +a. Since the positive number + coincides with the arithmetic number, then +a = a ∙, which means: we need to take three-quarters of the number a, i.e.

Therefore: the fractional positive coefficient shows how many times and what part of the letter factor of the monomial is repeated by the addend.

Polynomial should be easily represented in the form:

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 a monomial, or its certain fraction, taken with a minus sign, is repeated by the term.

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 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 to solve typical tasks with any monomials.

Subject:Monomials. Arithmetic operations on 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 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 appears 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 , , , .

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 appear only once in a monomial.

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:

1. 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.
2. Multiply all the same variables so that each variable appears only once in the monomial.
3. 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, it means 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 that 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.

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.

Lesson on the topic: "Standard form of a monomial. Definition. Examples"

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Monomial. Definition

Monomial- This mathematical expression, which is the product prime factor and one or more variables.

Monomials include all numbers, variables, their powers with a natural exponent:
42; 3; 0; 6 2 ; 2 3 ; b 3 ; ax 4 ; 4x 3 ; 5a 2 ; 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 this a monomial or not? To answer this question we need to simplify the expression, i.e. present in the form: $\frac(4)(5)*a^3$.
We can say for sure that this expression is a monomial.

Standard form of monomial

When performing calculations, it is advisable to reduce the monomial to standard form. This is the most concise and understandable recording of a monomial.

The procedure for reducing a monomial to standard form is as follows:
1. Multiply the coefficients of the monomial (or numerical factors) and place the resulting result in first place.
2. Select all powers 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.

Solution.
1. Multiply the coefficients of the monomial $15x^2y^3z * y^2z^4$.
2. Now we present similar terms $15x^2y^5z^5$.

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

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