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 standard operations on monomials, namely reduction to a standard form and calculation of a specific numerical value monomial at given values 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 over monomials
Lesson:The concept of a monomial. Standard view monomial
Consider some examples:
3. 
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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 called something like this algebraic expression, which 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 occurs 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:
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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»:
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So, here is a simplified expression:
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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. 
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3. 
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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 in this example monomial coefficient 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:
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2) multiply the powers:
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Let's write down the answer: ;
IN in this case the coefficient of the monomial is "", and the literal part 
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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, then we have arithmetic numeric expression, which should be calculated. 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 given example it is required to calculate the value of the monomial at , , , .
Monomials are products of numbers, variables and their powers. Numbers, variables and their powers 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. Coefficients 1 and -1 are not written, but a minus is kept 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. Numbers, variables and their powers are also considered monomials.
For example, the expressions 8, 35,y and y2 are monomials.
The standard form of a monomial is a monomial in the form of the product of a numerical factor in first place and powers of various variables. Any monomial can be reduced to a standard form by multiplying all the variables and numbers included in it. Here is an example of reducing a monomial to standard form:
4x2y4(-5)yx3 = 4(-5)x2x3y4y = -20x5y5
The numerical factor of a monomial written in standard form is called the coefficient of the monomial. For example, the coefficient of the monomial -7x2y2 is equal to -7. The coefficients of the 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 a 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 monomial 8x3yz2 is 6, the monomial 6x is 1, and the degree of -10 is 0.
Multiplying monomials. Raising monomials to powers
When multiplying monomials and raising monomials to powers, the rule of multiplying powers is used with the same basis and the rule for raising a degree to a degree. This produces a monomial, which is usually represented in standard form.
For example
4x3y2(-3)x2y = 4(-3)x3x2y2y = -12x5y3
((-5)x3y2)3 = (-5)3x3*3y2*3 = -125x9y6
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Lesson type: integrated (with ICT), lesson in introducing new knowledge.
Goals and objectives (algebra): introduce the concept of monomial; degree of monomial; standard form of monomial. Teach students to reduce monomials to standard form. Continue to develop skills in performing actions with degrees. Improve students' computing skills. Develop attentiveness and accuracy.
Goals and objectives (ICT): teach to use in practical activities built-in formula editor in MS Office Word; develop a skill independent work.
Materials used in the lesson: presentation, computer class with MS Office (Word) installed, reference summary practical work, task cards for independent work, multimedia installation.
During the classes
I. Organizational moment.
Greeting students.
II. Oral exercises.
(slide on screen2).
- Present as a power: y 3 *y 2 ; (y 3) 5 ; y 7 *y 3 ; (y 7) 4 ; a 10 /a 8 .
- What number (positive or negative) is the value of the expression: (-8) 10 ; (-5) 27 ; 7 5 ; -2 8 ; -(-1) 7 .
- Calculate: (3*2) 2 -3*2 2 ; (-3) 8 /3 7 .
III. Learning new material.
Reporting the topic of the lesson and the goals and objectives of the lesson (slide 3, 4).
6*x 2 *y; 2*x 3 ; mn 7; ab; -8 (slide 5)
- Read the expressions written on the board.
- What do these expressions represent?
Expressions of this type are called monomials.
DEFINITION: A monomial is the product of numbers and variables, powers of variables, or a number, variable, power of a variable.
Look carefully at the screen (slide 7). Which of the following expressions are monomials? Why?
IV. Consolidation of new material.
No. 463 – independently. Frontal check. (Slide 8).
V. Learning new material.
Let me have monomials
2x 2 y*9y 2 and 8x*9xy (slide 9)
Let's use the commutative and combinational laws multiplication. We get:
2*9*x 2 *y*y 2 =18x 2 y 3 and 8*9*x*x*y=72x 2 y.
- What did we get?
- What does it represent?
We represented the monomial as the product of the numerical factor in the first place and the powers of various variables. This type of monomial is called standard form.
- What monomial is called a monomial of standard form?
DEFINITION: a monomial is called a monomial of standard form if it has 1 numerical factor in the first place (coefficient), the product of identical variables in it is written as a power.
Read those monomials that are written in standard form. Name their coefficients.
VI. Consolidation of new material.
No. 464 - orally, No. 465 - under the guidance of a teacher.
VII. A task performed on a computer (practical work).
MS Word program. Built-in formula editor. Using the built-in formula editor to write monomials. File "Standard view of a monomial" on the desktop. Fill out the prepared table using the built-in formula editor.
Fill the table. (Slide 15)
Check - on the screen (slide 16) and saved student files.
VIII. Learning new material.
- What's written on the board?
- What is the exponent of the variable X?
- What is the exponent of the variable Y?
- Find the sum of the exponents. This number is called degree monomial.
On page 84 of the textbook, find the definition of the degree of a monomial. Read it.
IX. Consolidating new material.
No. 473 – orally;
No. 467 (a; d) - commented on the blackboard.
X. Independent work.
On the screen according to the options (slide 19). (Each student has a piece of paper on his desk with a task to complete the work - Appendix 2)
Check – self-test with recording (slide 20 on the screen).
XI. Summarizing.
- What is a monomial?
- What kind of monomial is it called? standard monomial?
- What is the degree of a monomial?
XII. Homework.
P.19, No. 466, 468, 476, 470.
Thank you for the lesson! (slide 23)
List of used literature:
- Algebra. 7th grade: textbook for educational institutions/ [Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov, S.B. Suvorov]; edited by S.A. Telyakovsky. - M.: Education, 2007.