The factorization of polynomials is identity transformation, as a result of which the polynomial is transformed into the product of several factors - polynomials or monomials.
There are several ways to factorize polynomials.
Method 1. Bracketing the common factor.
This transformation is based on the distributive law of multiplication: ac + bc = c(a + b). The essence of the transformation is to single out the common factor in the two components under consideration and “put it out” of the brackets.
Let us factorize the polynomial 28x 3 - 35x 4.
Decision.
1. We find a common divisor for elements 28x3 and 35x4. For 28 and 35 it will be 7; for x 3 and x 4 - x 3. In other words, our common factor is 7x3.
2. We represent each of the elements as a product of factors, one of which
7x 3: 28x 3 - 35x 4 \u003d 7x 3 ∙ 4 - 7x 3 ∙ 5x.
3. Bracketing the common factor
7x 3: 28x 3 - 35x 4 \u003d 7x 3 ∙ 4 - 7x 3 ∙ 5x \u003d 7x 3 (4 - 5x).
Method 2. Using abbreviated multiplication formulas. The "mastery" of mastering this method is to notice in the expression one of the formulas for abbreviated multiplication.
Let us factorize the polynomial x 6 - 1.
Decision.
1 TO given expression we can apply the difference of squares formula. To do this, we represent x 6 as (x 3) 2, and 1 as 1 2, i.e. 1. The expression will take the form:
(x 3) 2 - 1 \u003d (x 3 + 1) ∙ (x 3 - 1).
2. To the resulting expression, we can apply the formula for the sum and difference of cubes:
(x 3 + 1) ∙ (x 3 - 1) \u003d (x + 1) ∙ (x 2 - x + 1) ∙ (x - 1) ∙ (x 2 + x + 1).
So,
x 6 - 1 = (x 3) 2 - 1 = (x 3 + 1) ∙ (x 3 - 1) = (x + 1) ∙ (x 2 - x + 1) ∙ (x - 1) ∙ (x 2 + x + 1).
Method 3. Grouping. The grouping method consists in combining the components of a polynomial in such a way that it is easy to perform operations on them (addition, subtraction, taking out a common factor).
We factorize the polynomial x 3 - 3x 2 + 5x - 15.
Decision.
1. Group the components in this way: the 1st with the 2nd, and the 3rd with the 4th
(x 3 - 3x 2) + (5x - 15).
2. In the resulting expression, we take the common factors out of brackets: x 2 in the first case and 5 in the second.
(x 3 - 3x 2) + (5x - 15) \u003d x 2 (x - 3) + 5 (x - 3).
3. We take out the common factor x - 3 and get:
x 2 (x - 3) + 5 (x - 3) \u003d (x - 3) (x 2 + 5).
So,
x 3 - 3x 2 + 5x - 15 \u003d (x 3 - 3x 2) + (5x - 15) \u003d x 2 (x - 3) + 5 (x - 3) \u003d (x - 3) ∙ (x 2 + 5 ).
Let's fix the material.
Factor the polynomial a 2 - 7ab + 12b 2 .
Decision.
1. We represent the monomial 7ab as the sum 3ab + 4ab. The expression will take the form:
a 2 - (3ab + 4ab) + 12b 2 . 
Let's open the brackets and get:
a 2 - 3ab - 4ab + 12b 2 .
2. Group the components of the polynomial in this way: the 1st with the 2nd and the 3rd with the 4th. We get:
(a 2 - 3ab) - (4ab - 12b 2).
3. Let's take out the common factors:
(a 2 - 3ab) - (4ab - 12b 2) \u003d a (a - 3b) - 4b (a - 3b).
4. Let's take out the common factor (a - 3b):
a(a – 3b) – 4b(a – 3b) = (a – 3b) ∙ (a – 4b).
So,
a 2 - 7ab + 12b 2 =
= a 2 - (3ab + 4ab) + 12b 2 =
= a 2 - 3ab - 4ab + 12b 2 =
= (a 2 - 3ab) - (4ab - 12b 2) =
= a(a - 3b) - 4b(a - 3b) =
= (а – 3 b) ∙ (а – 4b).
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Public lesson
mathematics
in the 7th grade
"Application various ways for factoring a polynomial".
Prokofieva Natalya Viktorovna,
Mathematic teacher
Lesson Objectives
Educational:
- repeat abbreviated multiplication formulas
- formation and primary consolidation of the ability to factorize polynomials in various ways.
Developing:
- development of mindfulness, logical thinking, attention, the ability to systematize and apply the knowledge gained, mathematically literate speech.
Educational:
- formation of interest in solving examples;
- fostering a sense of mutual assistance, self-control, mathematical culture.
Lesson type: combined lesson
Equipment: projector, presentation, board, textbook.
Preliminary preparation for the lesson:
- Students should be familiar with the following topics:
- Squaring the sum and difference of two expressions
- Factoring with the squared sum and squared difference formulas
- Multiplying the difference of two expressions by their sum
- Factoring the difference of squares
- Factoring the sum and difference of cubes
- Be proficient in working with abbreviated multiplication formulas.
Lesson Plan
- Organizing time(direct students to the lesson)
- Examination homework(error correction)
- oral exercises
- Learning new material
- Training exercises
- repetition exercises
- Summing up the lesson
- Homework message
During the classes
I. Organizational moment.
The lesson will require you to know the formulas for abbreviated multiplication, the ability to apply them, and of course, attention.
II. Checking homework.
Questions on homework.
Debriefing on the board.
II. oral exercises.
Math is needed
It's impossible without her
We teach, we teach, friends,
What do we remember in the morning?
Let's do a workout.
Factorize (Slide 3)
8a-16b
17x² + 5x
c(x + y) + 5(x + y)
4a² - 25 (Slide 4)
1 - y³
ax + ay + 4x + 4y Slide 5)
III. Independent work.
Each of you has a table on the table. Sign your work at the top right. Fill in the table. The running time is 5 minutes. Started.
Finished.
Please swap jobs with a neighbor.
Put down your pens and grab your pencils.
We check the work - attention to the slide. (Slide 6)
We set the mark - (Slide 7)
7(+) - 5
6-5(+) - 4
4(+) - 3
Put the formulas in the middle of the table. Let's start learning new stuff.
IV. Learning new material
In notebooks we write down the number, class work and the topic of today's lesson.
Teacher.
- When factoring polynomials, sometimes not one, but several methods are used, applying them sequentially.
- Examples:
- 5a² - 20 \u003d 5 (a² - 4) \u003d 5 (a-2) (a + 2). (Slide 8)
We use the bracketing of the common factor and the difference of squares formula.
- 18x³ + 12x² + 2x = 2x (9x² + 6x + 1) = 2x (3x + 1)². (Slide 9)
What can be done with an expression? What method will we use to factorize?
Here we use the bracketing of the common factor and the square of the sum formula.
- ab³ - 3b³ + ab²y - 3b²y \u003d b² (ab - 3b + ay - 3y) \u003d b² ((ab - 3b) + (ay - 3y)) \u003d b² (b (a - 3) + y (a - 3)) \u003d b² (a - 3) (b + y). (Slide 10)
What can be done with an expression? What method will we use to factorize?
Here the common factor was taken out of brackets and the grouping method was applied.
- Factoring order: (Slide 11)
- Not every polynomial can be factorized. For example: x² + 1; 5x² + x + 2, etc. (Slide 12)
V. Training exercises
Before starting, we conduct a physical education minute (Slide 13)
They quickly got up and smiled.
Pulled higher and higher.
Come on, straighten your shoulders
Raise, lower.
Turn right, turn left
Sit down, get up. Sit down, get up.
And they ran on the spot.
And more gymnastics for the eyes:
- Close your eyes tightly for 3-5s, and then open them for 3-5s. We repeat 6 times.
- Place your thumb at a distance of 20-25 cm from the eyes, look with both eyes at the end of the finger for 3-5 seconds, and then look with both eyes at the pipe. We repeat 10 times.
Well done, have a seat.
Task for the lesson:
№934 avd
№935 av
№937
№939 avd
№1007 avd
VI. Exercises for repetition.
№ 933
VII. Summing up the lesson
The teacher asks questions, and the students answer them as they wish.
- Name the known methods of factoring a polynomial.
- Take the common factor out of the bracket
- Decomposition of a polynomial into factors using abbreviated multiplication formulas.
- grouping method
- Factoring order:
- Take the common factor out of the bracket (if any).
- Try to factorize the polynomial using the abbreviated multiplication formulas.
- If the previous methods did not lead to the goal, then try to apply the grouping method.
Raise your hand:
- If your attitude to the lesson is “I didn’t understand anything, and I didn’t succeed at all”
- If your attitude to the lesson is “there were difficulties, but I did it”
- If your attitude to the lesson “I did almost everything”
Factorize 4 a² - 25 = 1 - y³ = (2a - 5) (2a + 5) (1 - y) (1+y+y ²)
Factorize ax+ay+4x+4y= =a(x+y)+4(x+y)= (ax+ay)+(4x+4y)= (x+y) (a+4)
(a + b) ² a ² + 2ab + b ² Square of the sum a² - b² (a - b)(a + b) Difference of squares (a - b)² a² - 2ab + b² Square of the difference a³ + b ³ (a + b) (a² - ab + b²) Sum of cubes (a + b) ³ a³ + 3 a²b+3ab² + b³ Cube of sum (a - b) ³ a³ - 3a²b+3ab² - b³ Cube of difference a³ - b³ (a – b) (a² + ab + b²) Difference of cubes
MARKING 7 (+) = 5 6 or 5 (+) = 4 4 (+) = 3
Example #1. 5 a² - 20 = = 5(a² - 4) = = 5(a - 2) (a+2) Bracketing the common factor Difference of squares formula
Example #2. 18 x³ + 12x ² + 2x = =2x (9x ² +6x+1)= =2x(3x+1) ² Bracketing the common factor Sum formula
Example #3. ab³ –3b³+ab²y–3b²y= = b²(ab–3b+ay-3y)= =b²((a b -3 b)+(a y -3 y)= =b²(b(a-3)+y(a -3))= =b²(a-3)(b+y) Bracketize the factor Group the terms in brackets Bracket the factors Bracket the common factor
Factoring order Move the common factor out of the bracket (if any). Try to factorize the polynomial using the abbreviated multiplication formulas. 3. If the previous methods did not lead to the goal, then try to apply the grouping method.
Not every polynomial can be factorized. For example: x ² +1 5x ² + x + 2
PHYSICAL MINUTE
Assignment for lesson No. 934 ABD No. 935 ABD No. 937 No. 939 ABD No. 1007 ABD
Raise your hand: If your attitude to the lesson is “I didn’t understand anything, and I didn’t succeed at all” If your attitude to the lesson was “there were difficulties, but I did it” If your attitude to the lesson is “I did almost everything”
Homework: p. 38 No. 936 No. 938 No. 954
Sections: Mathematics
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: 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:
- Organizing time. 1 minute
- Formulating the topic, goals and objectives of the lesson-practice. 2 minutes
- Checking homework. 4 minutes
- Updating the basic knowledge and skills of students. 12 minutes
- Fizkultminutka. 2 minutes
- Instructions for completing the tasks of the workshop. 2 minutes
- Performing tasks in groups. 15 minutes
- Checking and discussing the performance of tasks. Work analysis. 3 minutes
- Setting homework. 1 minute
- 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 learning activities 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 the students at the blackboard to answer the questions that the students had in the field when checking on the 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:
- What does it mean to factor a polynomial?
- How many decomposition methods do you know?
- What are their names?
- 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), a sample solution for task No. 4 (done by the student) appears on the board.
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 the interactive whiteboard or a separate poster 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:
Governs 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
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) ; 4) . 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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Exist several different ways factorization of a polynomial. Most often, in practice, not one, but several methods are used at once. There can be no specific order of actions here, in each example everything is individual. But you can try to follow the following order:
1. If there is a common factor, then take it out of the bracket;
2. After that, try to factorize the polynomial using the abbreviated multiplication formulas;
3. If after that we have not yet received the desired result, we should try to use the grouping method.
Abbreviated multiplication formulas
1. a^2 - b^2 = (a+b)*(a-b);
2. (a+b)^2 = a^2+2*a*b+b^2;
3. (a-b)^2 = a^2-2*a*b+b^2;
4. a^3+b^3 = (a+b)*(a^2 - a*b+b^2);
5. a^3 - b^3 = (a-b)*(a^2 + a*b+b^2);
Now let's take a look at a few examples:
Example 1
Factorize the polynomial: (a^2+1)^2 - 4*a^2
First, we apply the abbreviated multiplication formula "difference of squares" and open the inner brackets.
(a^2+1)^2 - 4*a^2 = ((a^2+1)-2*a)*((a^2+1)+2*a) = (a^2+1 -2*a)*(a^2+1+2*a);
Note that the expressions for the square of the sum and the square of the difference of two expressions are obtained in brackets. Apply them and get the answer.
a^2+1-2*a)*(a^2+1+2*a) = (a-1)^2*(a+1)^2;
Answer:(a-1)^2*(a+1)^2;
Example 2
Factorize the polynomial 4*x^2 - y^2 + 4*x +2*y.
As you can see directly here, none of the methods is suitable. But there are two squares, they can be grouped. Let's try.
4*x^2 - y^2 + 4*x +2*y = (4*x^2 - y^2) +(4*x +2*y);
We got the formula for the difference of squares in the first bracket, And in the second bracket there is a common factor of two. Let's apply the formula and take out the common factor.
(4*x^2 - y^2) +(4*x +2*y)= (2*x - y)*(2*x+y) +2*(2*x+y);
It can be seen that two identical brackets are obtained. We take them out as a common factor.
(2*x - y)*(2*x+y) +2*(2*x+y) = (2*x+y)*(2*x - y)+2)= (2*x+ y)*(2*x-y+2);
Answer:(2*x+y)*(2*x-y+2);
As you can see, there is no universal way. With experience, the skill will come and factoring the polynomial into factors will be very easy.
