The concepts of "polynomial" and "factorization of a polynomial" in algebra are very common, because you need to know them in order to easily perform calculations with large multi-valued numbers. This article will describe several decomposition methods. All of them are quite simple to use, you just need to choose the right one in each case.

The concept of a polynomial

A polynomial is the sum of monomials, that is, expressions containing only the multiplication operation.

For example, 2 * x * y is a monomial, but 2 * x * y + 25 is a polynomial, which consists of 2 monomials: 2 * x * y and 25. Such polynomials are called binomials.

Sometimes, for the convenience of solving examples with multivalued values, the expression must be transformed, for example, decomposed into a certain number of factors, that is, numbers or expressions between which the multiplication operation is performed. There are a number of ways to factorize a polynomial. It is worth considering them starting from the most primitive, which is used even in primary classes.

Grouping (general entry)

The formula for factoring a polynomial into factors by the grouping method in general looks like this:

ac + bd + bc + ad = (ac + bc) + (ad + bd)

It is necessary to group the monomials so that a common factor appears in each group. In the first parenthesis, this is the factor c, and in the second - d. This must be done in order to then take it out of the bracket, thereby simplifying the calculations.

Decomposition algorithm on a specific example

The simplest example of factoring a polynomial into factors using the grouping method is given below:

10ac + 14bc - 25a - 35b = (10ac - 25a) + (14bc - 35b)

In the first bracket, you need to take the terms with the factor a, which will be common, and in the second - with the factor b. Pay attention to the + and - signs in the finished expression. We put before the monomial the sign that was in the initial expression. That is, you need to work not with the expression 25a, but with the expression -25. The minus sign, as it were, is “glued” to the expression behind it and always take it into account in calculations.

At the next step, you need to take out the factor, which is common, out of the bracket. That's what grouping is for. To take it out of the bracket means to write out before the bracket (omitting the multiplication sign) all those factors that are repeated exactly in all the terms that are in the bracket. If there are not 2, but 3 or more terms in the bracket, the common factor must be contained in each of them, otherwise it cannot be taken out of the bracket.

In our case, only 2 terms in brackets. The overall multiplier is immediately visible. The first parenthesis is a, the second is b. Here you need to pay attention to the digital coefficients. In the first bracket, both coefficients (10 and 25) are multiples of 5. This means that not only a, but also 5a can be bracketed. Before the bracket, write out 5a, and then divide each of the terms in brackets by the common factor that was taken out, and also write down the quotient in brackets, not forgetting the + and - signs. Do the same with the second bracket, take out 7b, since 14 and 35 multiple of 7.

10ac + 14bc - 25a - 35b = (10ac - 25a) + (14bc - 35b) = 5a (2c - 5) + 7b (2c - 5).

It turned out 2 terms: 5a (2c - 5) and 7b (2c - 5). Each of them contains a common factor (the whole expression in brackets here is the same, which means it is a common factor): 2c - 5. It also needs to be taken out of the bracket, that is, the terms 5a and 7b remain in the second bracket:

5a(2c - 5) + 7b(2c - 5) = (2c - 5)*(5a + 7b).

So the full expression is:

10ac + 14bc - 25a - 35b \u003d (10ac - 25a) + (14bc - 35b) \u003d 5a (2c - 5) + 7b (2c - 5) \u003d (2c - 5) * (5a + 7b).

Thus, the polynomial 10ac + 14bc - 25a - 35b is decomposed into 2 factors: (2c - 5) and (5a + 7b). The multiplication sign between them can be omitted when writing

Sometimes there are expressions of this type: 5a 2 + 50a 3, here you can bracket not only a or 5a, but even 5a 2. You should always try to take the largest possible common factor out of the bracket. In our case, if we divide each term by a common factor, we get:

5a 2 / 5a 2 = 1; 50a 3 / 5a 2 = 10a(when calculating the quotient of several powers with equal bases, the base is preserved, and the exponent is subtracted). Thus, one remains in the bracket (in no case do not forget to write one if you take out one of the terms entirely from the bracket) and the quotient of division: 10a. It turns out that:

5a 2 + 50a 3 = 5a 2 (1 + 10a)

Square formulas

For the convenience of calculations, several formulas have been derived. They are called reduced multiplication formulas and are used quite often. These formulas help factorize polynomials containing powers. This is another powerful way to factorize. So here they are:

• a 2 + 2ab + b 2 = (a + b) 2 - the formula, called the "square of the sum", since as a result of the expansion into a square, the sum of the numbers enclosed in brackets is taken, that is, the value of this sum is multiplied by itself 2 times, which means it is a multiplier.
• a 2 + 2ab - b 2 = (a - b) 2 - the formula of the square of the difference, it is similar to the previous one. The result is a difference enclosed in brackets, contained in a square power.
• a 2 - b 2 \u003d (a + b) (a - b)- this is the formula for the difference of squares, since initially the polynomial consists of 2 squares of numbers or expressions between which subtraction is performed. It is perhaps the most commonly used of the three.

Examples for calculating by formulas of squares

Calculations on them are made quite simply. For example:

1. 25x2 + 20xy + 4y 2 - use the formula "square of the sum".
2. 25x 2 is the square of 5x. 20xy is twice the product of 2*(5x*2y), and 4y 2 is the square of 2y.
3. So 25x 2 + 20xy + 4y 2 = (5x + 2y) 2 = (5x + 2y)(5x + 2y). This polynomial is decomposed into 2 factors (the factors are the same, therefore it is written as an expression with a square power).

Operations according to the formula of the square of the difference are performed similarly to these. What remains is the difference of squares formula. Examples for this formula are very easy to identify and find among other expressions. For example:

• 25a 2 - 400 \u003d (5a - 20) (5a + 20). Since 25a 2 \u003d (5a) 2, and 400 \u003d 20 2
• 36x 2 - 25y 2 \u003d (6x - 5y) (6x + 5y). Since 36x 2 \u003d (6x) 2, and 25y 2 \u003d (5y 2)
• c 2 - 169b 2 \u003d (c - 13b) (c + 13b). Since 169b 2 = (13b) 2

It is important that each of the terms is the square of some expression. Then this polynomial is to be factored by the difference of squares formula. For this, it is not necessary that the second power is above the number. There are polynomials containing large powers, but still suitable for these formulas.

a 8 +10a 4 +25 = (a 4) 2 + 2*a 4 *5 + 5 2 = (a 4 +5) 2

In this example, a 8 can be represented as (a 4) 2 , that is, the square of a certain expression. 25 is 5 2 and 10a is 4 - this is the double product of the terms 2*a 4 *5. That is, this expression, despite the presence of degrees with large exponents, can be decomposed into 2 factors in order to work with them later.

Cube formulas

The same formulas exist for factoring polynomials containing cubes. They are a little more complicated than those with squares:

• a 3 + b 3 \u003d (a + b) (a 2 - ab + b 2)- this formula is called the sum of cubes, since in its initial form the polynomial is the sum of two expressions or numbers enclosed in a cube.
• a 3 - b 3 \u003d (a - b) (a 2 + ab + b 2) - a formula identical to the previous one is denoted as the difference of cubes.
• a 3 + 3a 2 b + 3ab 2 + b 3 = (a + b) 3 - sum cube, as a result of calculations, the sum of numbers or expressions is obtained, enclosed in brackets and multiplied by itself 3 times, that is, located in the cube
• a 3 - 3a 2 b + 3ab 2 - b 3 = (a - b) 3 - the formula, compiled by analogy with the previous one with a change in only some signs of mathematical operations (plus and minus), is called the "difference cube".

The last two formulas are practically not used for the purpose of factoring a polynomial, since they are complex, and it is quite rare to find polynomials that completely correspond to just such a structure so that they can be decomposed according to these formulas. But you still need to know them, since they will be required for actions in the opposite direction - when opening brackets.

Examples for cube formulas

Consider an example: 64a 3 − 8b 3 = (4a) 3 − (2b) 3 = (4a − 2b)((4a) 2 + 4a*2b + (2b) 2) = (4a−2b)(16a 2 + 8ab + 4b 2 ).

We've taken fairly prime numbers here, so you can immediately see that 64a 3 is (4a) 3 and 8b 3 is (2b) 3 . Thus, this polynomial is expanded by the formula difference of cubes into 2 factors. Actions on the formula of the sum of cubes are performed by analogy.

It is important to understand that not all polynomials can be decomposed in at least one of the ways. But there are such expressions that contain larger powers than a square or a cube, but they can also be expanded into abbreviated multiplication forms. For example: x 12 + 125y 3 =(x 4) 3 +(5y) 3 =(x 4 +5y)*((x 4) 2 − x 4 *5y+(5y) 2)=(x 4 + 5y)( x 8 − 5x 4 y + 25y 2).

This example contains as many as 12 degrees. But even it can be factored using the sum of cubes formula. To do this, you need to represent x 12 as (x 4) 3, that is, as a cube of some expression. Now, instead of a, you need to substitute it in the formula. Well, the expression 125y 3 is the cube of 5y. The next step is to write the formula and do the calculations.

At first, or when in doubt, you can always check by inverse multiplication. You only need to open the brackets in the resulting expression and perform actions with similar terms. This method applies to all the listed methods of reduction: both to work with a common factor and grouping, and to operations on formulas of cubes and square powers.

The purpose of the lesson:  the formation of the skills of factoring a polynomial into factors in various ways;  to cultivate accuracy, perseverance, diligence, the ability to work in pairs. Equipment: multimedia projector, PC, didactic materials. Lesson plan: 1. Organizational moment; 2. Checking homework; 3. Oral work; 4. Learning new material; 5. Physical education; 6. Consolidation of the studied material; 7. Work in pairs; 8. Homework; 9. Summing up. Course of the lesson: 1. Organizational moment. Assign students to the lesson. Education does not consist in the amount of knowledge, but in the full understanding and skillful application of all that one knows. (Georg Hegel) 2. Checking homework. Analysis of tasks in the solution of which students had difficulties. 3. Oral work.  factorize: 1) 2) 3) ; four) .  Establish a correspondence between the expressions of the left and right columns: a. 1. b. 2. c. 3. d. 4. d. 5. .  Solve the equations: 1. 2. 3. 4. Learning new material. To factorize polynomials, we used parentheses, grouping, and abbreviated multiplication formulas. Sometimes it is possible to factorize a polynomial by applying successively several methods. You should start the transformation, if possible, by taking the common factor out of brackets. In order to successfully solve such examples, today we will try to develop a plan for their consistent application.

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LESSON PLAN algebra lesson in 7th grade

Teacher Prilepova O.A.

Lesson Objectives:

Show the application of various methods for factoring a polynomial

Repeat the methods of factorization and consolidate their knowledge during the exercises

To develop the skills and abilities of students in the application of abbreviated multiplication formulas.

Develop students' logical thinking and interest in the subject.

Tasks:

in the direction personal development:

Development of interest in mathematical creativity and mathematical abilities;

Development of initiative, activity in solving mathematical problems;

Cultivating the ability to make independent decisions.

in the meta-subject direction :

Formation of general ways of intellectual activity, characteristic of mathematics and being the basis of cognitive culture;

Use of ICT technology;

in the subject area:

Mastering the mathematical knowledge and skills necessary to continue education;

Formation in students the ability to look for ways to factorize a polynomial and find them for a polynomial that is factorized.

Equipment:handouts, route sheets with evaluation criteria,multimedia projector, presentation.

Lesson type:repetition, generalization and systematization of the material covered

Forms of work:work in pairs and groups, individual, collective,independent, frontal work.

During the classes:

Stages	Plan		UUD
Org moment.	Breakdown into groups and couples: Students choose a mate according to the following criterion: I communicate with this classmate the least. Psychological mood: Choose an emoticon of your choice (the mood at the beginning of the lesson) and under it look at the grade that you would like to receive today in the lesson (SLIDE). - Put yourself in the notebook in the margins of the grade that you would like to receive today in the lesson. You will mark your results in the table (SLIDE). Route sheet. Exercise total Grade Evaluation criteria: 1. I solved everything correctly, without errors - 5 2. When solving, I made from 1 to 2 mistakes - 4 3. Made 3 to 4 mistakes while solving - 3 4. Made more than 4 mistakes when solving - 2		New approaches to teaching (dialogue)
Actualization.	Collective work. - Today at the lesson you will be able to demonstrate your knowledge, participate in mutual control and self-control of your activities Match (SLIDE): On the next slide, pay attention to the expressions, what do you notice? (SLIDE) 15x3y2 + 5x2y Taking the common multiplier out of brackets p 2 + pq - 3 p -3 q Grouping method 16m2 - 4n2 Abbreviated multiplication formula How can these actions be united in one word? (Methods of expansion of polynomials) Statement by students of the topic and purpose of the lesson as their own learning task (SLIDE). Based on this, let's formulate the topic of our lesson and set goals. Questions for students: Name the topic of the lesson; Formulate the purpose of the lesson; Everyone has cards with the name of the formulas. (Work in pairs). Give formulas to all formulas
Application of knowledge	Work in pairs. Checking the slide 1. Choose the correct answer (SLIDE). Cards: Exercise Answer (x+10)2= x2+100-20x x2+100+20x x2+100+10x (5y-7)2= 25y2+49-70y 25u2-49-70u 25y2+49+70 x2-16y2= (x-4y)(x+4y) (x-16y)(x+16y) (x+4y)(4y-x) (2a+c)(2a-c)= 4a2-v2 4а2+в2 2a2-b2 a3-8v3 a2+16-64v6 (a-8c)(a+8c) (a-2c) (a2 + 2av + 4c2)	2. Find errors (SLIDE): Cards No. Checking the slide 1 pair: o ( b- y)2 = b2 - 4 by+y2 o 49- c2=(49-c)(49+s) 2 pair: o (r- 10) 2=r2- 20r+10 o (2a+1)2=4a2+2a+1 3 pair: o (3y+1)2=9y+6y+1 o ( b- a) 2 =b²- 4ba+a2 4 pair: o x²- 25= ( x-25)( 25+x) o (7- a) 2 \u003d 7- 14a + a²	Education in accordance with age characteristics
	3. Each pair is given tasks and a limited time to solve it (SLIDE) We check on the answer cards 1. Follow the steps: a) (a + 3c) 2; b) x 2 - 12 x + 36; c) 4v2-y2. 2. Factorize: a) ; b) ; in 2 x - a 2 y - 2 a 2 x + y 3. Find the value of the expression: (7 p + 4)2 -7 p (7 p - 2) at p = 5.		Management and leadership
	4. Group work. Look, make no mistake (SLIDE). Cards. Let's check the slide. (а+…)²=…+2…с+с² (... + y)² \u003d x² + 2x ... + ... (... + 2x)² \u003d y² + 4xy + 4x² (…+2 m)²=9+…+4 m² (n + 2v)²= n ²+…+4v²		Teaching critical thinking. Management and leadership
	5. Group work (consultation on the solution, discussion of tasks and their solutions) Each member of the group is given tasks of level A, B, C. Each member of the group chooses a feasible task for himself. Cards. (Slide) Checking with answer cards Level A 1. Factor it out: a) c 2 - a 2 ; b) 5x2-45; c) 5a2 + 10av + 5v2; d) ax2-4ax + 4a 2. Do the following: a) (x - 3) (x + 3); b) (x - 3)2; c) x (x - 4). Level B 1. Simplify: a) (3a + p) (3a-p) + p2; b) (a + 11) 2 - 20a; c) (a-4) (a + 4) -2a (3-a). 2. Calculate: a) 962 - 862; b) 1262 - 742. Level C 1. Solve the equation: (7 x - 8) (7x + 8) - (25x - 4)2 + 36(1 - 4x)2 =44 1. Solve the equation: (12 x - 4) (12 x + 4) - (12 x - 1)2 - (4 x - 5) = 16. 1.		Teaching the talented and gifted
Lesson summary	- Let's sum up, we will derive estimates according to the results of the table. Compare your scores with your estimated score. Choose the emoticon that matches your rating (SLIDE). c) the teacher evaluates the work of the class (activity, level of knowledge, skills, self-organization, diligence) Independent work in the form of a test with a RESERVE check		Assessment for Learning and Assessment for Learning
Homework	Continue teaching abbreviated multiplication formulas.
Reflection	Guys, please listen to the parable: (SLIDE) A sage was walking, and three people were meeting him, carrying carts with Stones for the construction of the Temple. The sage stopped and asked each Question. The first asked: - What did you do all day? And he replied with a smirk that he had been carrying cursed stones all day. The second asked: “And what did you do all day? ” And he replied: “I did my job conscientiously.” And the third smiled at him, his face lit up with joy and pleasure, and answered “A I took part in the construction of the Temple.” What is your Temple? (Knowledge) Guys! Who has worked since the first person? (show emoticons) (Score 3 or 2) (SLIDE) Who worked in good faith? (Score 4) And who took part in the construction of the Temple of Knowledge? (Score 5)		Critical Thinking Training

• Formation of skills to apply various methods for factorization.
• Contribute to the education of a culture of speech, accuracy of recording, independence.
• Formation of skills of partial search activity: to be aware of the problem, to analyze, to draw conclusions.

Equipment: textbook, blackboard, notebook, task cards.

Lesson type: Lesson of application of ZUN.

Teaching method: problematic, partially exploratory.

Form of organization of educational activities: group, frontal, individual, work in pairs.

Duration: 1 lesson (45 min)

Lesson plan:

1. Organization of the beginning of the lesson. (1 minute)
2. Checking homework. (2 minutes)
3. Actualization. (5 minutes)
4. Learning new material. (10 min)
5. Consolidation of new material. (15 minutes)
6. Control and self-examination of knowledge. (8 min)
7. Summarizing. (2 minutes)
8. Homework. (2 minutes)

During the classes

I. Organizational moment

Hello guys.

The topic of the lesson is “Application of various methods for factorization”. Today we will form the skills of using various methods of factorization and once again we will be convinced of the usefulness of the ability to factor a polynomial into factors.

I wish you to work actively in the lesson. (Write the topic in a notebook).

II. Checking homework

Before the start of the lesson, students hand in notebooks with completed homework for verification. Issues that caused difficulties are discussed.

III. Updating of basic knowledge.

Before we start solving problems, we will check how ready we are for this. Let's remember what we know about the topic of the lesson.

3.1. Front poll:

a) What does it mean to factor a polynomial?
b) What basic methods of factoring a polynomial do you know?
c) Any polynomial can be factorized? For example?
d) In what tasks is it sometimes useful to use factorization?

3.2. Draw lines to connect the polynomials with their corresponding factorization methods.

3.3. Find the wrong statement:

a) a 2 + b 2 - 2ab \u003d (a - b) 2

b) m 2 + 2mn - n 2 \u003d (m - n) 2

c) –2pt + p 2 + t 2 = (p - t) 2

d) 25 - 16 s 2 = (5 - 4s) (5 - 4s) (errors b, d)

3.4. Present as a product: a) 64x 2 - 1; b) (d - 3) 2 - 36;

3.5. Solve the Equation x 2 - 16 = 0 (4; -4)

3.5. Find the value of an expression 34 2 – 24 2 (580)

IV. Studying the material

To factorize polynomials, we used parentheses, grouping, and abbreviated multiplication formulas.

What do you think, are there situations in which it is possible to factorize a polynomial by applying successively several methods?

The following task will help us find the answer to this question:

Factor the polynomial and indicate which methods were used in this case. ( Work in pairs with the subsequent solution at the blackboard)

Example 1. 9x 3 - 36x used 2 methods:

Example 2. a 2 + 2ab + b 2 - c 2 used 2 methods:

• grouping;
• use of abbreviated multiplication formulas.

Example 3. y 3 - 3y 2 + 6y - 18 used 3 methods:

• grouping;
• use of abbreviated multiplication formulas;
• taking the common factor out of brackets.

Example 4. x 3 + 3x 2 + 2x used 3 ways:

• taking the common factor out of brackets;
• preliminary transformation;
• grouping.

We conclude: sometimes it is possible to factorize a polynomial by applying successively several methods. In order to successfully solve such examples, today let's develop a plan for consistently applying them:

1. Take the common factor out of the bracket (if any).
2. Try to factorize the polynomial using the abbreviated multiplication formulas.
3. Try to apply the grouping method (if the previous methods did not lead to the goal).

V. Exercises to consolidate the stated topic

5.1. The combination of various methods of factoring allows you to easily and gracefully perform arithmetic calculations, solve equations of the form ax 2 + bx + c \u003d 0 (a ≠ 0) (such equations are called quadratic, we will study them in grade 8).

* Solve the equation: a) x 2 - 17x + 72 = 0, b) x 2 + 10x + 21 = 0

Hint: Some term of the polynomial is decomposed into the necessary terms or supplemented by adding some term to it. In the latter case, so that the polynomial does not change, the same term is subtracted from it.

(Two students solve equations on their own in a notebook. Answer: a) 8; 9; b) - 1; - 5).

Complete the exercise from the textbook No. 1016 (c), 1017 (c), p. 186

(Two students decide on the board, the rest according to the options in the notebook).

5.2. Solve equations ( Pupils work in pairs, followed by self-examination)

No. 949, p.177 a) x 3 - x = 0 b) 9x - x 3 = 0 c) x 3 + x 2 = 0 d) 5x 4 - 2x 2 = 0

** (Individual tasks for more prepared students)

Card 1	Card 2	Card 3
Solve the equation and write the sum of the roots x 2 + 3x + 6 + 2x = 0		Solve the equation and write the sum of the roots

x(x+3) +2(3+x) =0 the sum is -5

The sum of the roots of this equation:

The sum of the roots of the equation:.

VI. Control and self-examination of knowledge.

The topic under consideration is an integral part of the GIA in mathematics. To control and self-test knowledge on this topic, you are invited to complete test tasks from the GIA training tasks. Circle your answer on the test questions.

Individual work on cards: (Students perform GIA test tasks, + self test)

Which of these expressions are identically equal to 4x-10y 2(2x-5y) -2(5y-2x) -10y-4x -10y+4x? a) 1; 3; b) all; c) 1;2;4; oppression	Which of these expressions are identically equal - 3 (-2a + y) -3(-y+2a) 6a-3y 3(2a-y) 3u-6a? and all; b) 2; y) 2;3; c)1;4	Which of these expressions are identically equal to -6a + 12p -6(a-2p) 12r-6a 6(-a+2p) -6(-p+a) ? a) 1; at all; c) 2;4; d)1;3
3a 3 -3a 2 -5a + 5. a) (a-1) (3a 2 +5); b) (a + 1) (3a 2 -5); c) (a-1) (5-3a 2); e) (a-1) (3a 2 +5).	Express as a product of polynomials 13ah-26x-5av + 10v. e) (a-2) (13x-5c); b) (a + 2) (3x-5c); c) (3a-6)(4x-c); d) (a-2) (5c-3x).	Express as a product of polynomials bу-6b-5у 2 +30у. a) (6-y) (b-5y); b) (y -6) (b + 5y); c) (y-6)(b-5y); d) (y -6) (5y - b).
Follow the steps: (5a-c) 2 . a) 25a 2 + 10ac + s 2; b) 25a 2 + 10ac-c 2; p) 25a 2 -10ac + c 2; d) 25a 2 -5ac + s 2.	Do the following: (5x + 2y) 2 . a) 25x 2 + 20xy + 4y 2; success

Teacher: Let's check the answers. Read the words you have. These are exactly the words that accompany seventh graders in preparation for the GIA in grade 9.

VII. Summing up the lesson

The teacher conducts a frontal review of the main stages of the lesson, evaluates the work of students and orients students in homework.

VIII. Homework: 38, No. 950 (p. 177), No. 1016 (g), 1017 (g), p. 186.

** Find the value of the expression (x+3)2 -2 (x+3) (x-3) +(x-3)2 at x=100.

The value of this expression does not depend on the choice of x.

The lesson is over. Thank you for the lesson and remember that knowledge that is not replenished daily decreases every day.

Used Books:

1. Textbook "Algebra Grade 7". Yu.N. Makarychev, N.G. Mindyuk and others. Ed. S.A. Telyakovsky. – M.; Enlightenment, 2009.
2. Collection of test tasks for thematic and final control. Algebra 7. I.L. Guseva and others - M.; Intellect Center, 2009.
3. State final certification (according to the new form): Grade 9. Thematic training tasks. Algebra / FIPI author-compiler: V.L. Kuznetsova. – M.: Eksmo, 2010.

Sections: Maths

Lesson type:

• according to the method of conducting - a practical lesson;
• for the didactic purpose - a lesson in the application of knowledge and skills.

Target: form the ability to factorize a polynomial.

Tasks:

• Didactic: to systematize, expand and deepen the knowledge, skills of students, apply various methods of factoring a polynomial into factors. To form the ability to apply the decomposition of a polynomial into factors by a combination of various techniques. To implement knowledge and skills on the topic: “Decomposition of a polynomial into factors” to complete tasks at a basic level and tasks of increased complexity.
• Educational: to develop mental activity through solving problems of various types, to learn to find and analyze the most rational ways of solving, to contribute to the formation of the ability to generalize the studied facts, to clearly and clearly express one's thoughts.
• Educational: develop skills of independent and team work, self-control skills.

Working methods:

• verbal;
• visual;
• practical.

Lesson equipment: interactive whiteboard or overhead scope, tables with abbreviated multiplication formulas, instructions, handout for group work.

Lesson structure:

1. Organizing time. 1 minute
2. Formulating the topic, goals and objectives of the lesson-practice. 2 minutes
3. Checking homework. 4 minutes
4. Updating the basic knowledge and skills of students. 12 minutes
5. Fizkultminutka. 2 minutes
6. Instructions for completing the tasks of the workshop. 2 minutes
7. Performing tasks in groups. 15 minutes
8. Checking and discussing the performance of tasks. Work analysis. 3 minutes
9. Setting homework. 1 minute
10. Reserve assignments. 3 minutes

During the classes

1. Organizational moment

The teacher checks the readiness of the classroom and students for the lesson.

2. Formulation of the topic, goals and objectives of the lesson-practice

• Message about the final lesson on the topic.
• Motivation of educational activity of students.
• Formulating the goal and setting the objectives of the lesson (together with students).

3. Checking homework

On the board are examples of solving homework exercises No. 943 (a, c); No. 945 (c, d). The samples were made by the students of the class. (This group of students was identified in the previous lesson, they formalized their decision at recess). The students prepare to “defend” the solutions.

Teacher:

Checks for homework in student notebooks.

Invites the students of the class to answer the question: “What difficulties did the assignment cause?”.

Offers to compare their solution with the solution on the board.

Invites students at the blackboard to answer questions that students had in the field when checking on samples.

He comments on the answers of students, supplements the answers, explains (if necessary).

Summarizes homework.

Students:

Present homework to the teacher.

Change notebooks (in pairs) and check with each other.

Answer the teacher's questions.

Check your solution with samples.

They act as opponents, make additions, corrections, write down a different method if the solution method in the notebook differs from the method on the board.

Ask for the necessary explanations to the students, to the teacher.

Find ways to check the results.

Participate in the assessment of the quality of the tasks at the blackboard.

4. Updating the basic knowledge and skills of students

1. Oral work

Teacher:

Answer the questions:

1. What does it mean to factor a polynomial?
2. How many decomposition methods do you know?
3. What are their names?
4. What is the most common?

2. Polynomials are written on the board:

1. 14x 3 - 14x 5

2. 16x 2 - (2 + x) 2

3. 9 - x 2 - 2xy - y 2

4.x3 - 3x - 2

Teacher invites students to factorize polynomials No. 1-3:

• Option I - by taking out a common factor;
• Option II - using abbreviated multiplication formulas;
• III variant - by way of grouping.

One student is offered to factorize the polynomial No. 4 (an individual task of increased difficulty, the task is performed on the A 4 format). Then a sample solution for tasks No. 1-3 (done by the teacher) appears on the board, a sample solution for task No. 4 (done by the student).

3. Warm up

The teacher gives instructions to factorize and choose the letter associated with the correct answer. By adding the letters you will get the name of the greatest mathematician of the 17th century, who made a huge contribution to the development of the theory of solving equations. (Descartes)

5. Physical education The students read the statements. If the statement is true, then the students should raise their hands up, and if it is not true, then sit down at the desk. (Annex 2)

6. Instruction on how to complete the tasks of the workshop.

On an interactive whiteboard or a separate poster, a table with instructions.

When decomposing a polynomial into factors, the following order must be observed:

1. put the common factor out of brackets (if any);

2. apply abbreviated multiplication formulas (if possible);

3. apply the grouping method;

4. check the result obtained by multiplication.

Teacher:

Offers instruction to students (emphasizes step 4).

Offers the implementation of workshop assignments in groups.

Distributes worksheets into groups, sheets with carbon paper for completing assignments in notebooks and their subsequent verification.

Determines the time for work in groups, for work in notebooks.

students:

They read the instructions.

Teachers listen carefully.

They are seated in groups (4-5 people each).

Prepare for practical work.

7. Performing tasks in groups

Worksheets with tasks for groups. (Annex 3)

Teacher:

Manages independent work in groups.

Evaluates the ability of students to work independently, the ability to work in a group, the quality of the design of the worksheet.

students:

Perform tasks on sheets of carbon paper enclosed in a workbook.

Discuss rational solutions.

Prepare a worksheet for the group.

Prepare to defend your work.

8. Checking and discussing the assignment

Answers on the whiteboard.

Teacher:

Collects copies of decisions.

Manages the work of students reporting on worksheets.

Offers to conduct a self-assessment of their work, compare answers in notebooks, worksheets and samples on the board.

Recalls the criteria for grading for work, for participation in its implementation.

Provides clarification on emerging decision or self-assessment issues.

Summarizes the first results of practical work and reflection.

Summarizes (together with students) the lesson.

Says that the final results will be summed up after checking copies of the work done by students.

students:

Give copies to the teacher.

Worksheets are attached to the board.

Reporting on the performance of work.

Perform self-assessment and self-assessment of work performance.

9. Setting homework

Homework is written on the board: No. 1016 (a, b); 1017 (c, d); No. 1021 (d, e, f)*

Teacher:

Offers to write down the obligatory part of the assignment at home.

Gives a comment on its implementation.

Invites more prepared students to write down No. 1021 (d, e, f) *.

Tells you to prepare for the next review review lesson