The concept of a monomial and its standard form. Reducing a monomial to a standard form, examples, solutions How to write a monomial in a standard form examples

In mathematics, there are many different mathematical expressions, and some of them have their own fixed names. We have to get acquainted with one of such concepts - it 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 monomials. As you can see, only one number or variable (with or without power) is also a monomial. But, for example, the expressions 2 + c, 3 * (y ^ 2) / x, a ^ 2 –x ^ 2 are already are not monomials, as they do not fit the definition. In the first expression, "sum" is used, which is unacceptable, in the second - "division", in the third - the difference.

Consider a few more examples.

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

Standard type 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 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 coefficient of the monomial.

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 the first place. Then multiply all the powers that have the same base letter.

Reduction of 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 reducing 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.

Monomial concept

Definition of a monomial: A monomial is an algebraic expression that uses only multiplication.

Standard type of monomial

What is the standard form of a monomial? A monomial is written in the standard form, if in it the first place is a numerical factor and this factor, it is called the coefficient of a monomial, 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 the standard form:

each letter occurs only once, they are located in the Latin alphabetical order, but where is the coefficient of the monomial, i.e. the numerical factor that should come first? It is here equal to one: 1adm.

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

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

If a 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 the standard form, for example: the number 5 is a monomial of the standard form.

Reducing monomials to standard form

How to bring a monomial to a standard form? Let's look at some examples.

Let the monomial 2a4b be given, you 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 the standard form. Multiply the numbers 2 and 4, replace the product aa with 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 these 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? Let us bring the second monomial to the standard form, we get 10abc. Now you 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 monomials? We can only summarize such monomials. Let's consider an 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 of the coefficients of the terms. So, the sum of 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. Only similar monomials can be added; addition is reduced to the addition of their coefficients.

Subtraction of monomials

What is the difference of monomials? We can subtract only similar monomials. Let's consider an example of subtraction of 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 monomials is 5a - 2a = 3a.

More examples of subtraction of 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

Multiplication of monomials

What is the product of monomials? Let's consider an example:

those. the product of monomials is equal to a monomial, the factors of which 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" to the degree: in the first - "a" to the degree of 2, and in the second - "a" to the degree 5. So the product will have "a" to the degree of 7, because when multiplying the same letters, the exponents of their degrees add 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.

These examples show that the coefficients of monomials are multiplied, and the same letters are replaced by the sums of their degrees in the product.

Purpose: - Get acquainted with the concept of a monomial;

Develop the ability to give examples of monomials

Determine if an expression is a monomial,

Indicate its coefficient and letter part.

Get to know the concept of "standard form of a monomial"

Introduce the algorithm for reducing the monomial to the standard form;

Develop practical skills in applying the algorithm

reducing the monomial to the standard form.

Download:

Preview:

To use preview presentations create yourself an account ( account) Google and log into it: https://accounts.google.com

Slide captions:

TOPIC: The concept of a monomial. Standard view of a monomial Purpose: - Get acquainted with the concept of a monomial; - Develop the ability to give examples of monomials - Determine whether an expression is a monomial, - Indicate its coefficient and letter part. -Get familiar with the concept of "standard form of a monomial" -Enter the algorithm for reducing a monomial to a standard form; To develop practical skills in applying the algorithm for reducing a monomial to a standard form.

A SINGLE TERM IS AN ALGEBRAIC EXPRESSION, WHICH IS THE PRODUCT OF NUMBERS AND VARIABLES INCREASED TO A DEGREE WITH A NATURAL INDICATOR. 2av, - 4a⁴v⁵, 1.7s⁸v⁴ 0; 2; -0.6; X; a; x ⁶ Not a monomial expression of the form: a + b; 2x⁴ + 3y⁹; a⁴⁄s ⁸ CONCEPT OF SINGLE TERM

Consider a monomial: 3a ∙ 4 a²b⁵c²bac⁵ = 3 ∙ 4aa²b⁵bc²c = 12a³b⁶c³ Mathematics strives for clarity, brevity and order. We have brought the monomial to a shorter notation i.e. to the standard view.

Algorithm. Bring the monomial to the standard form and name the coefficient of the monomial. 3х⁴ yz ∙ (-2) xy⁴z ⁸ = 3 ∙ (- 2) x⁴ ∙ х ∙ y⁴ ∙ y ∙ z ∙ z ⁸ = = -6х⁵ ∙ y⁵ ∙ z ⁹ ¼ab⁴c4c = ¼ ∙ 4ab⁴ (c ∙ c) = ab⁴c² ( 3/10) ab To bring a monomial to a standard form, you need to: 1) Multiply all numerical factors and put their product in the first place; 2) Multiply all available degrees with the same letter base; 3) Multiply all available degrees with another letter base, etc.

Bring the monomial to its standard form. Option 1 a) 7c⁴ · 4c³ · 8 c⁶ b) 8x² · 4 y³ · (- 2x ³) 2 version a) 6 n² · 3n³ · 9n⁶ b) 15 q⁴ · 2p² · (-5p⁵)

Let's check the answers of independent work. 1 option a) 244 c¹³ b) -64 x ⁸ y³ 2 option a) 162 n ¹¹ b) - 150 q ⁴ p⁷

On the subject: methodological developments, presentations and notes

Presentation on mathematics on the topic "The concept of a monomial. Standard form of a monomial". The presentation has been compiled for review new topic in mathematics in grade 7 "The concept of a monomial. The standard form of a monomial ...

the concept of a monomial. standard kind of monomial

presentation for an algebra lesson in grade 7 on the topic "The concept of a monomial. Standard form of a monomial". the concepts of a monomial, a degree of a monomial, a coefficient of a monomial, a standard form of a monomial are given ...

I. Expressions that are composed of numbers, variables and their powers, using the multiplication action, are called monomials.

Examples of monomials:

a) a; b) ab; v) 12; G)-3c; e) 2a 2 ∙ (-3.5b) 3; e)-123.45xy 5 z; g) 8ac ∙ 2.5a 2 ∙ (-3c 3).

II. This type of monomial, when in the first place there is a numerical factor (coefficient), followed by the variables with their degrees, is called the standard form of the monomial.

So, the monomials given above, under the letters a B C), G) and e) are written in the standard form, and the monomials under the letters e) and g) it is required to bring it to the standard form, that is, to such a form when the numerical factor is in the first place, followed by the letter factors with their indicators, and the letter factors are in alphabetical order. Here are the monomials e) and g) to the standard view.

e) 2a 2 ∙ (-3.5b) 3= 2a 2 ∙ (-3.5) 3 ∙ b 3 = -2a 2 ∙ 3.5 ∙ 3.5 ∙ 3.5 ∙ b 3 = -85.75a 2 b 3;

g) 8ac ∙ 2.5a 2 ∙ (-3c 3)= -8 ∙ 2.5 ∙ 3a 3 c 3 = -60a 3 s 3.

III.The sum of the exponents of the degrees of all the variables that make up the monomial is called the degree of the monomial.

Examples. What degree do the monomials have? a) - g)?

a) a. First;

b) ab. Second: a in the first degree and b in the first degree - the sum of indicators 1+1=2 ;

v) 12. Zero, since there are no alphabetic factors;

G) -3c. First;

e) -85.75a 2 b 3. Fifth. We have reduced this monomial to the standard form, we have a in the second degree and b in the third. Add up the indicators: 2+3=5 ;

e) -123.45xy 5 z. Seventh. We added the exponents of the letter multipliers: 1+5+1=7 ;

g) -60a 3 s 3. Sixth, since the sum of the alphabetic multipliers 3+3=6 .

IV. Monomials that have the same letter part are called similar monomials.

Example. Indicate similar monomials among these monomials 1) -7).

1) 3aabbc; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 2 bac; 5) 10aaa 2 x; 6) -2.3a 4 x; 7) 34x 2 y.

Here are the monomials 1), 4) and 5) to the standard view. Then the line of these monomials will look like this:

1) 3a 2 b 2 c; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 3 bc; 5) 10a 4 x; 6) -2.3a 4 x; 7) 34x 2 y.

Similar will be those that have the same letter part, i.e. 1) and 3); 2) and 4); 5) and 6).

1) 3a 2 b 2 c and 3) 56a 2 b 2 c;

2) -4.1a 3 bc and 4) 98.7a 3 bc;

5) 10a 4 x and 6) -2.3a 4 x.

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

Additional materials
Dear users, do not forget to leave your comments, reviews, wishes. All materials have been checked by an antivirus program.

Teaching aids and simulators in the Integral online store for grade 7
Electronic study guide "Clear geometry" for grades 7-9
Multimedia study guide "Geometry in 10 minutes" for grades 7-9

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 degrees with a natural exponent:
42; 3; 0; 6 2; 2 3; b 3; ax 4; 4x 3; 5a 2; 12xyz 3.

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

Standard type of monomial

When calculating, it is desirable to bring the monomial to a standard form. This is the most concise and understandable notation for a monomial.

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

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

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

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

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