Ministry of Education of the Republic of Belarus
Educational institution
"Gomel State University them. F. Skorina"
Faculty of Mathematics
Department of MPM
Identical transformations of expressions and methods of teaching students how to perform them
Executor:
Student Starodubova A.Yu.
Cand. physics and mathematics Sciences, Associate Professor Lebedeva M.T.
Gomel 2007
Introduction
1 The main types of transformations and stages of their study. Stages of mastering the use of transformations
Conclusion
Literature
Introduction
The simplest transformations of expressions and formulas, based on the properties of arithmetic operations, are carried out in primary school and 5th and 6th grades. The formation of skills and abilities to perform transformations takes place in an algebra course. This is due both to the sharp increase in the number and variety of transformations being carried out, and to the complication of activities to justify them and clarify the conditions of applicability, to the identification and study of the generalized concepts of identity, identical transformation, equivalent transformation.
1. Main types of transformations and stages of their study. Stages of mastering the use of transformations
1. Beginnings of algebra
An undivided system of transformations is used, represented by rules for performing actions on one or both parts of the formula. The goal is to achieve fluency in completing tasks for solving simple equations, simplifying formulas that define functions, and rationally carrying out calculations based on the properties of actions.
Typical examples:
Solve equations:
A) ; b) ; V) .
Identical transformation (a); equivalent and identical (b).
2. Formation of skills in applying specific types of transformations
Conclusions: abbreviated multiplication formulas; transformations associated with exponentiation; transformations associated with various classes of elementary functions.
Organization whole system transformations (synthesis)
The goal is to create a flexible and powerful device suitable for use in solving a variety of educational assignments . The transition to this stage is carried out during the final repetition of the course in the course of understanding the known material learned in parts, by certain types transformations add transformations of trigonometric expressions to the previously studied types. All these transformations can be called “algebraic”; “analytical” transformations include those that are based on the rules of differentiation and integration and transformation of expressions containing passages to limits. The difference of this type is in the nature of the set that the variables in identities (certain sets of functions) run through.
The identities being studied are divided into two classes:
I – identities of abbreviated multiplication valid in a commutative ring and identities
fair in the field.
II – identities connecting arithmetic operations and basic elementary functions.
2 Features of the organization of the system of tasks when studying identity transformations
The main principle of organizing the system of tasks is to present them from simple to complex.
Exercise cycle– combining in a sequence of exercises several aspects of studying and techniques for arranging the material. When studying identity transformations, a cycle of exercises is associated with the study of one identity, around which other identities that are in a natural connection with it are grouped. The cycle, along with executive ones, includes tasks, requiring recognition of the applicability of the identity in question. The identity under study is used to carry out calculations on various numerical domains. The tasks in each cycle are divided into two groups. TO first These include tasks performed during initial acquaintance with identity. They serve educational material for several consecutive lessons united by one topic.
Second group exercises connects the identity being studied with various applications. This group does not form a compositional unity - the exercises here are scattered on various topics.
The described cycle structures refer to the stage of developing skills for applying specific transformations.
At the stage of synthesis, the cycles change, groups of tasks are combined in the direction of complication and merging of cycles related to various identities, which helps to increase the role of actions to recognize the applicability of a particular identity.
Example.
Cycle of tasks for identity:
I group of tasks:
a) present in the form of a product:
b) Check the equality:
c) Expand the parentheses in the expression:
.
d) Calculate:
e) Factorize:
f) simplify the expression:
.
Students have just become familiar with the formulation of an identity, its writing in the form of an identity, and its proof.
Task a) is associated with fixing the structure of the identity being studied, with establishing a connection with numerical sets(comparison of sign structures of identity and transformed expression; replacement of a letter with a number in an identity). IN last example it is still necessary to reduce it to the species being studied. In the following examples (e and g) there is a complication caused by applied role identity and complication of the sign structure.
Tasks of type b) are aimed at developing replacement skills 
on . The role of task c) is similar.
Examples of type d), in which it is necessary to choose one of the directions of transformation, complete the development of this idea.
Group I tasks are focused on mastering the structure of an identity, the operation of substitution in the simplest, fundamentally most important cases, and the idea of the reversibility of transformations carried out by an identity. Very important has also enrichment linguistic means showing various aspects identities. The texts of the assignments give an idea of these aspects.
II group of tasks.
g) Using the identity for , factor the polynomial .
h) Eliminate irrationality in the denominator of the fraction.
i) Prove that if - odd number, then it is divisible by 4.
j) The function is given analytical expression
.
Get rid of the modulus sign by considering two cases: , .
k) Solve the equation 
.
These tasks are aimed at as much as possible full use and taking into account the specifics of this particular identity, presuppose the formation of skills in using the identity being studied for the difference of squares. The goal is to deepen the understanding of identity by considering its various applications in different situations, combined with the use of material related to other topics in the mathematics course.
or 
.
Features of task cycles related to identities for elementary functions:
1) they are studied on the basis of functional material;
2) the identities of the first group appear later and are studied using already developed skills for carrying out identity transformations.
The first group of tasks in the cycle should include tasks to establish connections between these new numerical areas and the original area of rational numbers.
Example.
Calculate:
;
.
The purpose of such tasks is to master the features of records, including symbols of new operations and functions, and to develop mathematical speech skills.
Much of the use of identity transformations associated with elementary functions, falls on the solution of irrational and transcendental equations. Sequence of steps:
a) find the function φ for which given equation f(x)=0 can be represented as:
b) substitute y=φ(x) and solve the equation
c) solve each of the equations φ(x)=y k, where y k is the set of roots of the equation F(y)=0.
When using the described method, step b) is often performed implicitly, without introducing a notation for φ(x). In addition, students often prefer different ways leading to finding the answer, choose the one that leads to the algebraic equation faster and easier.
Example. Solve the equation 4 x -3*2=0.
2)(2 2) x -3*2 x =0 (step a)
(2 x) 2 -3*2 x =0; 2 x (2 x -3)=0; 2 x -3=0. (step b)
Example. Solve the equation:
a) 2 2x -3*2 x +2=0;
b) 2 2x -3*2 x -4=0;
c) 2 2x -3*2 x +1=0.
(Suggest for independent solution.)
Classification of tasks in cycles related to the solution of transcendental equations, including exponential function:
1) equations that reduce to equations of the form a x =y 0 and have a simple, general answer:
2) equations that reduce to equations of the form a x = a k, where k is an integer, or a x = b, where b≤0.
3) equations that reduce to equations of the form a x =y 0 and require explicit analysis form in which the number y 0 is explicitly written.
Tasks in which identity transformations are used to construct graphs when simplifying formulas that define functions.
a) Graph the function y=;
b) Solve the equation lgx+lg(x-3)=1
c) on what set is the formula log(x-5)+ log(x+5)= log(x 2 -25) an identity?
The use of identity transformations in calculations. (Journal of Mathematics at School, No. 4, 1983, p. 45)
Task No. 1. The function is given by the formula y=0.3x 2 +4.64x-6. Find the values of the function at x=1.2
y(1,2)=0.3*1.2 2 +4.64*1.2-6=1.2(0.3*1.2+4.64)-6=1.2(0 .36+4.64)-6=1.2*5-6=0.
Task No. 2. Calculate leg length right triangle, if the length of its hypotenuse is 3.6 cm, and the other leg is 2.16 cm.
Task No. 3. What is the area of the plot rectangular shape, having dimensions a) 0.64 m and 6.25 m; b) 99.8m and 2.6m?
a)0.64*6.25=0.8 2 *2.5 2 =(0.8*2.5) 2;
b)99.8*2.6=(100-0.2)2.6=100*2.6-0.2*2.6=260-0.52.
These examples make it possible to identify practical use identity transformations. The student should be familiarized with the conditions for the feasibility of the transformation (see diagrams).
| |
- image of a polynomial, where any polynomial fits into round contours. (Diagram 1)
-
the condition for the feasibility of transforming the product of a monomial and an expression that allows transformation into a difference of squares is given. (scheme 2)
-
here the shadings mean equal monomials and an expression is given that can be converted into a difference of squares. (Scheme 3)
| |
| |
- an expression that allows for a common factor.
Students’ skills in identifying conditions can be developed using the following examples:
Which of the following expressions can be transformed by taking the common factor out of brackets:
2) 
3) 0.7a 2 +0.2b 2 ;
5) 6,3*0,4+3,4*6,3;
6) 2x 2 +3x 2 +5y 2 ;
7) 0,21+0,22+0,23.
Most calculations in practice do not satisfy the conditions of satisfiability, so students need the skills to reduce them to a form that allows calculation of transformations. In this case, the following tasks are appropriate:
when studying taking the common factor out of brackets:
convert this expression, if possible, into an expression that is depicted in diagram 4:
4) 2a*a 2 *a 2;
5) 2n 4 +3n 6 +n 9 ;
8) 15ab 2 +5a 2 b;
10) 12,4*-1,24*0,7;
11) 4,9*3,5+1,7*10,5;
12) 10,8 2 -108;
13) 
14) 5*2 2 +7*2 3 -11*2 4 ;
15) 2*3 4 -3*2 4 +6;
18) 3,2/0,7-1,8*
When forming the concept of “identical transformation”, it should be remembered that this means not only that the given and the resulting expression as a result of the transformation take on equal values for any values of the letters included in it, but also that during the identical transformation we move from the expression that defines one way of calculating to an expression defining another way of calculating the same value.
Scheme 5 (the rule for converting the product of a monomial and a polynomial) can be illustrated with examples
0.5a(b+c) or 3.8(0.7+).
Exercises to learn how to take a common factor out of brackets:
Calculate the value of the expression:
a) 4.59*0.25+1.27*0.25+2.3-0.25;
b) a+bc at a=0.96; b=4.8; c=9.8.
c) a(a+c)-c(a+b) with a=1.4; b=2.8; c=5.2.
Let us illustrate with examples the formation of skills in calculations and identity transformations. (Journal of Mathematics at School, No. 5, 1984, p. 30)
1) skills and abilities are acquired faster and retained longer if their formation occurs on a conscious basis ( didactic principle consciousness).
1) You can formulate a rule for adding fractions with like denominators or preliminarily specific examples consider the essence of adding equal shares.
2) When factoring by taking the common factor out of brackets, it is important to see this common multiplier and then apply the distributive law. When performing the first exercises, it is useful to write each term of the polynomial as a product, one of the factors of which is common to all terms:
3a 3 -15a 2 b+5ab 2 = a3a 2 -a15ab+a5b 2 .
It is especially useful to do this when one of the monomials of a polynomial is taken out of brackets:
II. First stage formation of a skill - mastery of a skill (exercises are performed with detailed explanations and records)
(the issue of the sign is resolved first)
Second phase– the stage of automating the skill by eliminating some intermediate operations
III. Strength of skills is achieved by solving examples that are varied in both content and form.
Topic: “Putting the common factor out of brackets.”
1. Write down the missing factor instead of the polynomial:
2. Factorize so that before the brackets there is a monomial with a negative coefficient:
3. Factor so that the polynomial in brackets has integer coefficients:
4. Solve the equation:
IV. Skill development is most effective when some intermediate calculations or transformations are performed orally.
(orally);
V. The skills and abilities being developed must be part of the previously formed system of knowledge, skills and abilities of students.
For example, when teaching how to factor polynomials using abbreviated multiplication formulas, the following exercises are offered:
Factorize:
VI. The need for rational execution of calculations and transformations.
V) simplify the expression:
Rationality lies in opening the parentheses, because
VII. Converting expressions containing exponents.
No. 1011 (Alg.9) Simplify the expression:
No. 1012 (Alg.9) Remove the multiplier from under the root sign:
No. 1013 (Alg.9) Enter a factor under the root sign:
No. 1014 (Alg.9) Simplify the expression:
In all examples, first perform either factorization, or subtraction of the common factor, or “see” the corresponding reduction formula.
No. 1015 (Alg.9) Reduce the fraction:
Many students experience some difficulty in transforming expressions containing roots, in particular when studying equality:
Therefore, either describe in detail expressions of the form or 
or go to a degree with a rational exponent.
No. 1018 (Alg.9) Find the value of the expression:
No. 1019 (Alg.9) Simplify the expression:
2.285 (Skanavi) Simplify the expression
and then plot the function y For
No. 2.299 (Skanavi) Check the validity of the equality:
Transformation of expressions containing a degree is a generalization of acquired skills and abilities in the study of identical transformations of polynomials.
No. 2.320 (Skanavi) Simplify the expression:
The Algebra 7 course provides the following definitions.
Def. Two expressions whose corresponding values are equal for the values of the variables are said to be identically equal.
Def. Equality is true for any values of the variables called. identity.
No. 94 (Alg.7) Is the equality:
a) 
c) 
d) 
Description definition: Replacing one expression with another identically equal expression is called an identical transformation or simply a transformation of an expression. Identical transformations of expressions with variables are performed based on the properties of operations on numbers.
No. (Alg.7) Among the expressions
find those that are identically equal.
Topic: “Identical transformations of expressions” (question technique)
The first topic of “Algebra-7” - “Expressions and their transformations” helps to consolidate the computational skills acquired in grades 5-6, systematize and generalize information about transformations of expressions and solutions to equations.
Finding the values of numeric and literal expressions makes it possible to repeat with students the rules of action with rational numbers. The ability to perform arithmetic operations with rational numbers is fundamental to the entire algebra course.
When considering transformations of expressions, formal and operational skills remain at the same level that was achieved in grades 5-6.
However, here students rise to a new level in mastering theory. The concepts of “identically equal expressions”, “identity”, “identical transformations of expressions” are introduced, the content of which will constantly be revealed and deepened when studying the transformations of various algebraic expressions. It is emphasized that the basis of identity transformations is the properties of operations on numbers.
When studying the topic “Polynomials”, formal operational skills of identical transformations of algebraic expressions are formed. Abbreviated multiplication formulas contribute to the further process of developing the ability to perform identical transformations of whole expressions; the ability to apply formulas for both abbreviated multiplication and factorization of polynomials is used not only in transforming whole expressions, but also in operations with fractions, roots, powers with a rational exponent .
In the 8th grade, the acquired skills of identity transformations are practiced in actions with algebraic fractions, square root and expressions containing powers with an integer exponent.
In the future, the techniques of identity transformations are reflected in expressions containing a degree with a rational exponent.
A special group of identity transformations consists of trigonometric expressions and logarithmic expressions.
Mandatory learning outcomes for an algebra course in grades 7-9 include:
1) identity transformations of integer expressions
a) opening and enclosing brackets;
b) bringing similar members;
c) addition, subtraction and multiplication of polynomials;
d) factoring polynomials by putting the common factor out of brackets and abbreviated multiplication formulas;
e) decomposition quadratic trinomial by multipliers.
“Mathematics at school” (B.U.M.) p.110
2) identity transformations rational expressions: addition, subtraction, multiplication and division of fractions, as well as apply the listed skills when performing simple combined transformations [p. 111]
3) students should be able to perform transformations of simple expressions containing powers and roots. (pp. 111-112)
The main types of problems were considered, the ability to solve which allows the student to receive a positive grade.
One of the most important aspects of the methodology for studying identity transformations is the student’s development of goals for performing identity transformations.
1) 
- simplification numerical value expressions
2) which of the transformations should be performed: (1) 
or (2) Analysis of these options is a motivation (preferable (1), since in (2) the scope of definition is narrowed)
3) Solve the equation:
Factoring when solving equations.
4) Calculate:
Let's apply the abbreviated multiplication formula:
(101-1) (101+1)=100102=102000
5) Find the value of the expression:
To find the value, multiply each fraction by its conjugate:
6) Graph the function:
Let's select the whole part: .
Prevention of errors when performing identity transformations can be obtained by varying examples of their implementation. In this case, “small” techniques are practiced, which, as components, are included in a larger transformation process.
For example:
Depending on the directions of the equation, several problems can be considered: multiplication of polynomials from right to left; from left to right - factorization. Left side is a multiple of one of the factors on the right side, etc.
In addition to varying the examples, you can use apologia between identities and numerical equalities.
The next technique is the explanation of identities.
To increase students' interest, we can include finding in various ways problem solving.
Lessons on studying identity transformations will become more interesting if you devote them to searching for a solution to the problem .
For example: 1) reduce the fraction:
3) prove the formula of the “complex radical”
Consider:
Let's transform the right side of the equality:
-
the sum of conjugate expressions. They could be multiplied and divided by their conjugate, but such an operation would lead us to a fraction whose denominator is the difference of the radicals.
Note that the first term in the first part of the identity is a number greater than the second, so we can square both parts:
Practical lesson №3.
Topic: Identical transformations of expressions (question technique).
Literature: “Workshop on MPM”, pp. 87-93.
Sign high culture calculations and identity transformations for students are solid knowledge properties and algorithms of operations on exact and approximate quantities and their skillful application; rational techniques calculations and transformations and their verification; the ability to justify the use of methods and rules of calculations and transformations, automatic skills of error-free execution of computational operations.
At what grade should students begin working on developing the listed skills?
The line of identical transformations of expressions begins with the use of techniques rational calculation begins with the use of techniques for rationally calculating the values of numerical expressions. (5th grade)
When studying such topics school course mathematics should be given to them Special attention!
Students' conscious implementation of identity transformations is facilitated by the understanding of the fact that algebraic expressions do not exist on their own, but in inextricable connection with a certain numerical set, they are generalized records of numerical expressions. Analogies between algebraic and numerical expressions (and their transformations) are logical; their use in teaching helps prevent students from making mistakes.
Identical transformations are not a separate topic in the school mathematics course; they are studied throughout the entire course of algebra and the beginnings of mathematical analysis.
The mathematics program for grades 1-5 is propaedeutic material for studying identical transformations of expressions with a variable.
In the 7th grade algebra course. the definition of identity and identity transformations is introduced.
Def. Two expressions whose corresponding values are equal for any values of the variables are called. identically equal.
ODA. An equality that is true for any values of the variables is called an identity.
The value of identity lies in the fact that it allows a given expression to be replaced by another that is identically equal to it.
Def. Replacing one expression with another identically equal expression is called identical transformation or simply transformation expressions.
Identical transformations of expressions with variables are performed based on the properties of operations on numbers.
The basis of identity transformations can be considered equivalent transformations.
ODA. Two sentences, each of which is a logical consequence of the other, are called. equivalent.
ODA. Sentence with variables A is called. consequence of a sentence with variables B, if the domain of truth B is a subset of the domain of truth A.
Another definition of equivalent sentences can be given: two sentences with variables are equivalent if their truth domains coincide.
a) B: x-1=0 over R; A: (x-1) 2 over R => A~B, because areas of truth (solution) coincide (x=1)
b) A: x=2 over R; B: x 2 =4 over R => domain of truth A: x = 2; truth domain B: x=-2, x=2; because the domain of truth of A is contained in B, then: x 2 =4 is a consequence of the proposition x = 2.
The basis of identity transformations is the ability to represent the same number in different forms. For example,
-
This representation will help when studying the topic “basic properties of fractions.”
Skills in performing identity transformations begin to develop when solving examples similar to the following: “Find the numerical value of the expression 2a 3 +3ab+b 2 with a = 0.5, b = 2/3,” which are offered to students in grade 5 and allow for propaedeutics concept of function.
When studying abbreviated multiplication formulas, you should pay attention to their deep understanding and strong assimilation. To do this, you can use the following graphic illustration:
| |
(a+b) 2 =a 2 +2ab+b 2 (a-b) 2 =a 2 -2ab+b 2 a 2 -b 2 =(a-b)(a+b)
Question: How to explain to students the essence of the given formulas based on these drawings?
A common mistake is to confuse the expressions “square of the sum” and “sum of squares.” The teacher's indication that these expressions differ in the order of operation does not seem significant, since students believe that these actions are performed on the same numbers and therefore the result does not change by changing the order of actions.
Assignment: Create oral exercises to develop students’ skills in using the above formulas without errors. How can we explain how these two expressions are similar and how they differ from each other?
The wide variety of identical transformations makes it difficult for students to orient themselves as to the purpose for which they are performed. Fuzzy knowledge of the purpose of performing transformations (in each specific case) negatively affects their awareness, serves as a source massive errors students. This suggests that explaining to students the goals of performing various identity transformations is important. integral part methods for studying them.
Examples of motivations for identity transformations:
1. simplification of location numerical value expressions;
2. choosing a transformation of the equation that does not lead to the loss of the root;
3. When performing a transformation, you can mark its calculation area;
4. use of transformations in calculations, for example, 99 2 -1=(99-1)(99+1);
To manage the decision process, it is important for the teacher to have the ability to give an accurate description of the essence of the mistake made by the student. Accurate error characterization is key to the right choice subsequent actions taken by the teacher.
Examples of student errors:
1. performing multiplication: the student received -54abx 6 (7 cells);
2. By raising to a power (3x 2) 3 the student received 3x 6 (7 grades);
3. transforming (m + n) 2 into a polynomial, the student received m 2 + n 2 (7th grade);
4. By reducing the fraction the student received (8 grades);
5. performing subtraction: 
, student writes down (8th grade)
6. Representing the fraction in the form of fractions, the student received: 
(8 grades);
7. Removing arithmetic root the student received x-1 (grade 9);
8. solving the equation (9th grade);
9. By transforming the expression, the student receives: (9th grade).
Conclusion
The study of identity transformations is carried out in close connection with numerical sets studied in a particular class.
At first, you should ask the student to explain each step of the transformation, to formulate the rules and laws that apply.
In identical transformations of algebraic expressions, two rules are used: substitution and replacement by equals. Substitution is most often used, because Calculation using formulas is based on it, i.e. find the value of the expression a*b with a=5 and b=-3. Very often, students neglect parentheses when performing multiplication operations, believing that the multiplication sign is implied. For example, the following entry is possible: 5*-3.
Literature
1. A.I. Azarov, S.A. Barvenov “Functional and graphic methods solving exam problems”, Mn..Aversev, 2004
2. O.N. Piryutko “Typical mistakes in centralized testing", Mn..Aversev, 2006
3. A.I. Azarov, S.A. Barvenov “Trap tasks in centralized testing”, Mn..Aversev, 2006
4. A.I. Azarov, S.A. Barvenov “Methods of solution trigonometric problems", Mn..Aversev, 2005
Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:
\(5a^4 - 2a^3 + 0.3a^2 - 4.6a + 8\)
\(xy^3 - 5x^2y + 9x^3 - 7y^2 + 6x + 5y - 2\)
The sum of monomials is called a polynomial. The terms in a polynomial are called terms of the polynomial. Monomials are also classified as polynomials, considering a monomial to be a polynomial consisting of one member.
For example, a polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.
Let's represent all terms in the form of monomials standard view:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16\)
Let us present similar terms in the resulting polynomial:
\(8b^5 -14b^5 +3b^2 -8b -3b^2 + 16 = -6b^5 -8b + 16 \)
The result is a polynomial, all terms of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.
Behind degree of polynomial of a standard form take the highest of the powers of its members. Thus, the binomial \(12a^2b - 7b\) has the third degree, and the trinomial \(2b^2 -7b + 6\) has the second.
Typically, the terms of standard form polynomials containing one variable are arranged in descending order of exponents. For example:
\(5x - 18x^3 + 1 + x^5 = x^5 - 18x^3 + 5x + 1\)
The sum of several polynomials can be transformed (simplified) into a polynomial of standard form.
Sometimes the terms of a polynomial need to be divided into groups, enclosing each group in parentheses. Since enclosing parentheses is the inverse transformation of opening parentheses, it is easy to formulate rules for opening brackets:
If a “+” sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.
If a “-” sign is placed before the brackets, then the terms enclosed in the brackets are written with opposite signs.
Transformation (simplification) of the product of a monomial and a polynomial
Using the distributive property of multiplication, you can transform (simplify) the product of a monomial and a polynomial into a polynomial. For example:
\(9a^2b(7a^2 - 5ab - 4b^2) = \)
\(= 9a^2b \cdot 7a^2 + 9a^2b \cdot (-5ab) + 9a^2b \cdot (-4b^2) = \)
\(= 63a^4b - 45a^3b^2 - 36a^2b^3 \)
The product of a monomial and a polynomial is identically equal to the sum of the products of this monomial and each of the terms of the polynomial.
This result is usually formulated as a rule.
To multiply a monomial by a polynomial, you must multiply that monomial by each of the terms of the polynomial.
We have already used this rule several times to multiply by a sum.
Product of polynomials. Transformation (simplification) of the product of two polynomials
In general, the product of two polynomials is identically equal to the sum of the product of each term of one polynomial and each term of the other.
Usually the following rule is used.
To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other and add the resulting products.
Abbreviated multiplication formulas. Sum squares, differences and difference of squares
With some expressions in algebraic transformations have to deal with more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), i.e. the square of the sum, the square of the difference and difference of squares. You noticed that the names of these expressions seem to be incomplete, for example, \((a + b)^2 \) is, of course, not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b does not occur very often; as a rule, instead of the letters a and b, it contains various, sometimes quite complex, expressions.
The expressions \((a + b)^2, \; (a - b)^2 \) can be easily converted (simplified) into polynomials of the standard form; in fact, you have already encountered this task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)
It is useful to remember the resulting identities and apply them without intermediate calculations. Brief verbal formulations help this.
\((a + b)^2 = a^2 + b^2 + 2ab \) - square of the sum equal to the sum squares and double the product.
\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is equal to the sum of squares without the doubled product.
\(a^2 - b^2 = (a - b)(a + b) \) - the difference of squares is equal to the product of the difference and the sum.
These three identities allow one to replace its left-hand parts with right-hand ones in transformations and vice versa - right-hand parts with left-hand ones. The most difficult thing is to see the corresponding expressions and understand how the variables a and b are replaced in them. Let's look at several examples of using abbreviated multiplication formulas.
Important notes!
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We often hear this unpleasant phrase: “simplify the expression.” Usually we see some kind of monster like this:
“It’s much simpler,” we say, but such an answer usually doesn’t work.
Now I will teach you not to be afraid of any such tasks.
Moreover, at the end of the lesson you will simplify this example to (just!) regular number(yes, to hell with these letters).
But before you start this activity, you need to be able to handle fractions And factor polynomials.
Therefore, if you have not done this before, be sure to master the topics “” and “”.
Have you read it? If yes, then you are now ready.
Let's go! (Let's go!)
Basic Expression Simplification Operations
Now let's look at the basic techniques that are used to simplify expressions.
The simplest one is
1. Bringing similar
What are similar? You took this in 7th grade, when letters instead of numbers first appeared in mathematics.
Similar- these are terms (monomials) with the same letter part.
For example, in total similar terms- this is i.
Do you remember?
Give similar- means adding several similar terms to each other and getting one term.
How can we put the letters together? - you ask.
This is very easy to understand if you imagine that the letters are some kind of objects.
For example, a letter is a chair. Then what is the expression equal to?
Two chairs plus three chairs, how many will it be? That's right, chairs: .
Now try this expression: .
To avoid confusion, let different letters represent different objects.
For example, - is (as usual) a chair, and - is a table.
chairs tables chair tables chairs chairs tables
The numbers by which the letters in such terms are multiplied are called coefficients.
For example, in a monomial the coefficient is equal. And in it is equal.
So, the rule for bringing similar ones is:
Examples:
Give similar ones:
Answers:
2. (and similar, since, therefore, these terms have the same letter part).
2. Factorization
This is usually the most an important part in simplifying expressions.
After you have given similar ones, most often the resulting expression is needed factorize, that is, presented in the form of a product.
Especially this important in fractions: after all, in order to be able to reduce the fraction, The numerator and denominator must be represented as a product.
You went through the methods of factoring expressions in detail in the topic “”, so here you just have to remember what you learned.
To do this, solve several examples (you need to factorize them)
Examples:
Solutions:
3. Reducing a fraction.
Well, what could be more pleasant than crossing out part of the numerator and denominator and throwing them out of your life?
That's the beauty of downsizing.
It's simple:
If the numerator and denominator contain the same factors, they can be reduced, that is, removed from the fraction.
This rule follows from the basic property of a fraction:
That is, the essence of the reduction operation is that We divide the numerator and denominator of the fraction by the same number (or by the same expression).
To reduce a fraction you need:
1) numerator and denominator factorize
2) if the numerator and denominator contain common factors, they can be crossed out.
Examples:
The principle, I think, is clear?
I would like to draw your attention to one thing typical mistake when contracting. Although this topic is simple, many people do everything wrong, not understanding that reduce- this means divide numerator and denominator are the same number.
No abbreviations if the numerator or denominator is a sum.
For example: we need to simplify.
Some people do this: which is absolutely wrong.
Another example: reduce.
The “smartest” will do this:
Tell me what's wrong here? It would seem: - this is a multiplier, which means it can be reduced.
But no: - this is a factor of only one term in the numerator, but the numerator itself as a whole is not factorized.
Here's another example: .
This expression is factorized, which means you can reduce it, that is, divide the numerator and denominator by, and then by:
You can immediately divide it into:
To avoid such mistakes, remember easy way how to determine whether an expression is factorized:
The arithmetic operation that is performed last when calculating the value of an expression is the “master” operation.
That is, if you substitute some (any) numbers instead of letters and try to calculate the value of the expression, then if last action there will be a multiplication, which means we have a product (the expression is factorized).
If the last action is addition or subtraction, this means that the expression is not factorized (and therefore cannot be reduced).
To reinforce this, solve a few examples yourself:
Examples:
Solutions:
4. Adding and subtracting fractions. Reducing fractions to a common denominator.
Addition and subtraction ordinary fractions- the operation is well known: we look for a common denominator, multiply each fraction by the missing factor and add/subtract the numerators.
Let's remember:
Answers:
1. The denominators and are relatively prime, that is, they do not have common factors. Therefore, the LCM of these numbers is equal to their product. This will be the common denominator:
2. Here the common denominator is:
3. First thing here mixed fractions we turn them into incorrect ones, and then follow the usual pattern:
It's a completely different matter if the fractions contain letters, for example:
Let's start with something simple:
a) Denominators do not contain letters
Everything here is the same as with ordinary numerical fractions: find the common denominator, multiply each fraction by the missing factor and add/subtract the numerators:
Now in the numerator you can give similar ones, if any, and factor them:
Try it yourself:
Answers:
b) Denominators contain letters
Let's remember the principle of finding a common denominator without letters:
· first of all, we determine the common factors;
· then we write out all the common factors one at a time;
· and multiply them by all other non-common factors.
To determine the common factors of the denominators, we first factor them into prime factors:
Let us emphasize the common factors:
Now let’s write out the common factors one at a time and add to them all the non-common (not underlined) factors:
This is the common denominator.
Let's get back to the letters. The denominators are given in exactly the same way:
· factor the denominators;
· determine common (identical) factors;
· write out all common factors once;
· multiply them by all other non-common factors.
So, in order:
1) factor the denominators:
2) determine common (identical) factors:
3) write out all the common factors once and multiply them by all other (non-underlined) factors:
So there's a common denominator here. The first fraction must be multiplied by, the second - by:
By the way, there is one trick:
For example: .
We see the same factors in the denominators, only all with different indicators. The common denominator will be:
to a degree
to a degree
to a degree
to a degree.
Let's complicate the task:
How to make fractions have the same denominator?
Let's remember the basic property of a fraction:
Nowhere does it say that the same number can be subtracted (or added) from the numerator and denominator of a fraction. Because it's not true!
See for yourself: take any fraction, for example, and add some number to the numerator and denominator, for example, . What did you learn?
So, another unshakable rule:
When you reduce fractions to common denominator, use only the multiplication operation!
But what do you need to multiply by to get?
So multiply by. And multiply by:
We will call expressions that cannot be factorized “elementary factors.”
For example, - this is an elementary factor. - Same. But no: it can be factorized.
What about the expression? Is it elementary?
No, because it can be factorized:
(you already read about factorization in the topic “”).
So, the elementary factors into which you expand the expression with letters are an analogue prime factors, into which you decompose the numbers. And we will deal with them in the same way.
We see that both denominators have a multiplier. It will go to the common denominator to the degree (remember why?).
The factor is elementary, and they do not have a common factor, which means that the first fraction will simply have to be multiplied by it:
Another example:
Solution:
Before you multiply these denominators in a panic, you need to think about how to factor them? They both represent:
Great! Then:
Another example:
Solution:
As usual, let's factorize the denominators. In the first denominator we simply put it out of brackets; in the second - the difference of squares:
It would seem that there are no common factors. But if you look closely, they are similar... And it’s true:
So let's write:
That is, it turned out like this: inside the bracket we swapped the terms, and at the same time the sign in front of the fraction changed to the opposite. Take note, you will have to do this often.
Now let's bring it to a common denominator:
Got it? Let's check it now.
Tasks for independent solution:
Answers:
5. Multiplication and division of fractions.
Well, the hardest part is over now. And ahead of us is the simplest, but at the same time the most important:
Procedure
What is the procedure for calculating a numerical expression? Remember by calculating the meaning of this expression:
Did you count?
It should work.
So, let me remind you.
The first step is to calculate the degree.
The second is multiplication and division. If there are several multiplications and divisions at the same time, they can be done in any order.
And finally, we perform addition and subtraction. Again, in any order.
But: the expression in brackets is evaluated out of turn!
If several brackets are multiplied or divided by each other, we first calculate the expression in each of the brackets, and then multiply or divide them.
What if there are more brackets inside the brackets? Well, let's think: some expression is written inside the brackets. When calculating an expression, what should you do first? That's right, calculate the brackets. Well, we figured it out: first we calculate the inner brackets, then everything else.
So, the procedure for the expression above is as follows (the current action is highlighted in red, that is, the action that I am performing right now):
Okay, it's all simple.
But this is not the same as an expression with letters?
No, it's the same! Only instead of arithmetic operations you need to do algebraic, that is, the actions described in the previous section: bringing similar, adding fractions, reducing fractions, and so on. The only difference will be the action of factoring polynomials (we often use this when working with fractions). Most often, to factorize, you need to use I or simply put the common factor out of brackets.
Usually our goal is to represent the expression as a product or quotient.
For example:
Let's simplify the expression.
1) First, we simplify the expression in brackets. There we have a difference of fractions, and our goal is to present it as a product or quotient. So, we bring the fractions to a common denominator and add:
It is impossible to simplify this expression any further; all the factors here are elementary (do you still remember what this means?).
2) We get:
Multiplying fractions: what could be simpler.
3) Now you can shorten:
OK it's all over Now. Nothing complicated, right?
Another example:
Simplify the expression.
First, try to solve it yourself, and only then look at the solution.
Solution:
First of all, let's determine the order of actions.
First, let's add the fractions in parentheses, so instead of two fractions we get one.
Then we will do division of fractions. Well, let's add the result with the last fraction.
I will number the steps schematically:
Finally I'll give you two useful advice:
1. If there are similar ones, they must be brought immediately. At whatever point similar ones arise in our country, it is advisable to bring them up immediately.
2. The same applies to reducing fractions: as soon as the opportunity to reduce appears, it must be taken advantage of. The exception is for fractions that you add or subtract: if they now have same denominators, then the reduction should be left for later.
Here are some tasks for you to solve on your own:
And what was promised at the very beginning:
Answers:
Solutions (brief):
If you have coped with at least the first three examples, then you have mastered the topic.
Now on to learning!
CONVERTING EXPRESSIONS. SUMMARY AND BASIC FORMULAS
Basic simplification operations:
- Bringing similar: to add (reduce) similar terms, you need to add their coefficients and assign the letter part.
- Factorization: putting the common factor out of brackets, applying it, etc.
- Reducing a fraction: The numerator and denominator of a fraction can be multiplied or divided by the same non-zero number, which does not change the value of the fraction.
1) numerator and denominator factorize
2) if the numerator and denominator have common factors, they can be crossed out.IMPORTANT: only multipliers can be reduced!
- Adding and subtracting fractions:
; - Multiplying and dividing fractions:
;
Well, the topic is over. If you are reading these lines, it means you are very cool.
Because only 5% of people are able to master something on their own. And if you read to the end, then you are in this 5%!
Now the most important thing.
You have understood the theory on this topic. And, I repeat, this... this is just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough...
For what?
For successful passing the Unified State Exam, for admission to college on a budget and, MOST IMPORTANTLY, for life.
I won’t convince you of anything, I’ll just say one thing...
People who received a good education, earn much more than those who did not receive it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because there is much more open before them more possibilities and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the Unified State Exam and ultimately be... happier?
GAIN YOUR HAND BY SOLVING PROBLEMS ON THIS TOPIC.
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You will need solve problems against time.
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It's like in sports - you need to repeat it many times to win for sure.
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Basic properties of addition and multiplication of numbers.
Commutative property of addition: rearranging the terms does not change the value of the sum. For any numbers a and b the equality is true
Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third to the first number. For any numbers a, b and c the equality is true
Commutative property of multiplication: rearranging the factors does not change the value of the product. For any numbers a, b and c the equality is true
Combinative property of multiplication: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third.
For any numbers a, b and c the equality is true
Distributive Property: To multiply a number by a sum, you can multiply that number by each term and add the results. For any numbers a, b and c the equality is true
From the commutative and combinative properties of addition it follows: in any sum you can rearrange the terms in any way you like and arbitrarily combine them into groups.
Example 1 Let's calculate the sum 1.23+13.5+4.27.
To do this, it is convenient to combine the first term with the third. We get:
1,23+13,5+4,27=(1,23+4,27)+13,5=5,5+13,5=19.
From the commutative and combinative properties of multiplication it follows: in any product you can rearrange the factors in any way and arbitrarily combine them into groups.
Example 2 Let's find the value of the product 1.8·0.25·64·0.5.
Combining the first factor with the fourth, and the second with the third, we have:
1.8·0.25·64·0.5=(1.8·0.5)·(0.25·64)=0.9·16=14.4.
The distributive property is also true when a number is multiplied by the sum of three or more terms.
For example, for any numbers a, b, c and d the equality is true
a(b+c+d)=ab+ac+ad.
We know that subtraction can be replaced by addition by adding to the minuend the opposite number of the subtrahend:
This allows numeric expression type a-b be considered the sum of numbers a and -b, a numerical expression of the form a+b-c-d be considered the sum of numbers a, b, -c, -d, etc. The considered properties of actions are also valid for such sums.
Example 3 Let's find the value of the expression 3.27-6.5-2.5+1.73.
This expression is the sum of the numbers 3.27, -6.5, -2.5 and 1.73. Applying the properties of addition, we get: 3.27-6.5-2.5+1.73=(3.27+1.73)+(-6.5-2.5)=5+(-9) = -4.
Example 4 Let's calculate the product 36·().
The multiplier can be thought of as the sum of the numbers and -. Using the distributive property of multiplication, we obtain:
36()=36·-36·=9-10=-1.
Identities
Definition. Two expressions whose corresponding values are equal for any values of the variables are called identically equal.
Definition. An equality that is true for any values of the variables is called an identity.
Let's find the values of the expressions 3(x+y) and 3x+3y for x=5, y=4:
3(x+y)=3(5+4)=3 9=27,
3x+3y=3·5+3·4=15+12=27.
We got the same result. From the distribution property it follows that, in general, for any values of the variables, the corresponding values of the expressions 3(x+y) and 3x+3y are equal.
Let us now consider the expressions 2x+y and 2xy. When x=1, y=2 they take equal values:
However, you can specify values of x and y such that the values of these expressions are not equal. For example, if x=3, y=4, then
The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal.
The equality 3(x+y)=x+3y, true for any values of x and y, is an identity.
True numerical equalities are also considered identities.
Thus, identities are equalities that express the basic properties of operations on numbers:
a+b=b+a, (a+b)+c=a+(b+c),
ab=ba, (ab)c=a(bc), a(b+c)=ab+ac.
Other examples of identities can be given:
a+0=a, a+(-a)=0, a-b=a+(-b),
a·1=a, a·(-b)=-ab, (-a)(-b)=ab.
Identical transformations of expressions
Replacing one expression with another identically equal expression is called an identical transformation or simply a transformation of an expression.
Identical transformations of expressions with variables are performed based on the properties of operations on numbers.
To find the value of the expression xy-xz when given values x, y, z, you need to perform three actions. For example, with x=2.3, y=0.8, z=0.2 we get:
xy-xz=2.3·0.8-2.3·0.2=1.84-0.46=1.38.
This result can be obtained by performing only two steps, if you use the expression x(y-z), which is identically equal to the expression xy-xz:
xy-xz=2.3(0.8-0.2)=2.3·0.6=1.38.
We simplified the calculations by replacing the expression xy-xz identically equal expression x(y-z).
Identical transformations of expressions are widely used in calculating the values of expressions and solving other problems. Some identical transformations have already had to be performed, for example, bringing similar terms, opening parentheses. Let us recall the rules for performing these transformations:
to bring similar terms, you need to add their coefficients and multiply the result by the common letter part;
if there is a plus sign before the brackets, then the brackets can be omitted, preserving the sign of each term enclosed in brackets;
If there is a minus sign before the parentheses, then the parentheses can be omitted by changing the sign of each term enclosed in the parentheses.
Example 1 Let us present similar terms in the sum 5x+2x-3x.
Let's use the rule for reducing similar terms:
5x+2x-3x=(5+2-3)x=4x.
This transformation is based on the distributive property of multiplication.
Example 2 Let's open the brackets in the expression 2a+(b-3c).
Using the rule for opening parentheses preceded by a plus sign:
2a+(b-3c)=2a+b-3c.
The transformation carried out is based on associative property addition.
Example 3 Let's open the brackets in the expression a-(4b-c).
Let's use the rule for opening parentheses preceded by a minus sign:
a-(4b-c)=a-4b+c.
The transformation performed is based on the distributive property of multiplication and the combinatory property of addition. Let's show it. Let's imagine in this expression the second term -(4b-c) in the form of a product (-1)(4b-c):
a-(4b-c)=a+(-1)(4b-c).
By applying specified properties actions, we get:
a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.
The numbers and expressions that make up the original expression can be replaced by identically equal expressions. Such a transformation of the original expression leads to an expression that is identically equal to it.
For example, in the expression 3+x, the number 3 can be replaced by the sum 1+2, which will result in the expression (1+2)+x, which is identically equal to the original expression. Another example: in the expression 1+a 5, the power a 5 can be replaced by an identically equal product, for example, of the form a·a 4. This will give us the expression 1+a·a 4 .
This transformation is undoubtedly artificial, and is usually a preparation for some further transformations. For example, in the sum 4 x 3 +2 x 2, taking into account the properties of the degree, the term 4 x 3 can be represented as a product 2 x 2 2 x. After this transformation, the original expression will take the form 2 x 2 2 x+2 x 2. Obviously, the terms in the resulting sum have a common factor of 2 x 2, so we can perform the following transformation - bracketing. After it we come to the expression: 2 x 2 (2 x+1) .
Adding and subtracting the same number
Another artificial transformation of an expression is the addition and simultaneous subtraction of the same number or expression. This transformation is identical because it is essentially equivalent to adding zero, and adding zero does not change the value.
Let's look at an example. Let's take the expression x 2 +2·x. If you add one to it and subtract one, this will allow you to perform another identical transformation in the future - square the binomial: x 2 +2 x=x 2 +2 x+1−1=(x+1) 2 −1.
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.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- 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.