Numerical and algebraic expressions. Converting Expressions. Online calculator. Simplifying a polynomial. Multiplying polynomials
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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 yourself will simplify this example to (just!) an ordinary 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 the sum, similar terms are and.
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 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 a 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 decompose an expression with letters are an analogue of the simple factors into which you decompose 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 ones, 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 will give you two useful tips:
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 completion 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 many more opportunities open up before them 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.
You won't be asked for theory during the exam.
You will need solve problems against time.
And, if you haven’t solved them (A LOT!), you’ll definitely make a stupid mistake somewhere or simply won’t have time.
It's like in sports - you need to repeat it many times to win for sure.
Find the collection wherever you want, necessarily with solutions, detailed analysis and decide, decide, decide!
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Find problems and solve them!
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.
Topic No. 2.
Converting algebraic expressions
I. Theoretical material
Basic Concepts
Algebraic expression: integer, fractional, rational, irrational.
Scope of definition, valid expression values.
The meaning of an algebraic expression.
Monomial, polynomial.
Abbreviated multiplication formulas.
Factorization, putting the common factor out of brackets.
The main property of a fraction.
Degree, properties of degree.
Kortym, properties of roots.
Transformation of rational and irrational expressions.
An expression made up of numbers and variables using the signs of addition, subtraction, multiplication, division, raising to a rational power, extracting the root and using parentheses is called algebraic.
For example: ; 
; 
;
; 
; 
; 
.
If an algebraic expression does not contain division into variables and taking the root of variables (in particular, raising to a power with a fractional exponent), then it is called whole.
For example: 
; 
; 
.
If an algebraic expression is composed of numbers and variables using the operations of addition, subtraction, multiplication, exponentiation with natural indicator and division, and division into expressions with variables is used, then it is called fractional.
For example: 
; 
.
Whole and fractional expressions are called rational expressions.
For example: ; 
;
.
If an algebraic expression involves taking the root of variables (or raising variables to a fractional power), then such an algebraic expression is called irrational.
For example: 
; 
.
The values of the variables for which the algebraic expression makes sense are called valid variable values.
The set of all possible values of variables is called domain of definition.
The domain of definition of an entire algebraic expression is the set of real numbers.
The domain of definition of a fractional algebraic expression is the set of all real numbers except those that make the denominator zero.
For example: makes sense when 
;
makes sense when 
, that is, when 
.
The domain of definition of an irrational algebraic expression is the set of all real numbers, except those that turn into a negative number the expression under the sign of the root of an even power or under the sign of raising to a fractional power.
For example: 
makes sense when 
;
makes sense when 
, that is, when 
.
The numerical value obtained by substituting the permissible values of variables into an algebraic expression is called the value of an algebraic expression.
For example: expression 
at 
, 
takes on the value 
.
An algebraic expression containing only numbers, natural powers of variables and their products is called monomial.
For example: 
; 
; 
.
The monomial, written as the product of the numerical factor in the first place and the powers of various variables, is reduced to standard view.
For example: 
; 
.
The numerical factor of the standard notation of a monomial is called coefficient of the monomial. The sum of the exponents of all variables is called degree of monomial.
When multiplying a monomial by a monomial and when raising a monomial to natural degree we get a monomial that needs to be brought to standard form.
The sum of monomials is called polynomial.
For example: 
; ; 
.
If all members of a polynomial are written in standard form and similar members are reduced, then the resulting polynomial of standard form.
For example: .
If there is only one variable in a polynomial, then the largest exponent of this variable is called degree of polynomial.
For example: A polynomial has the fifth degree.
The value of the variable at which the value of the polynomial is zero is called root of the polynomial.
For example: roots of a polynomial 
are the numbers 1.5 and 2.
Abbreviated multiplication formulas
Special cases of using abbreviated multiplication formulas
Difference of squares: 
or
Squared sum: 
or
Squared difference: 
or
Sum of cubes: 
or
Difference of cubes: 
or
Cube of sum: 
or
Difference cube: 
or
Converting a polynomial into a product of several factors (polynomials or monomials) is called factoring a polynomial.
For example:.
Methods for factoring a polynomial
For example: .
Using abbreviated multiplication formulas.
For example: .
Grouping method. Commutative and associative laws allow you to group members of a polynomial different ways. One of the methods leads to the fact that the same expression is obtained in brackets, which in turn is taken out of brackets.
For example:.
Any fractional algebraic expression can be written as the quotient of two rational expressions with a variable in the denominator.
For example: 
.
A fraction in which the numerator and denominator are rational expressions and the denominator has a variable is called rational fraction.
For example: 
; 
; 
.
If the numerator and denominator of a rational fraction are multiplied or divided by the same nonzero number, monomial, or polynomial, the value of the fraction does not change. This expression is called the main property of a fraction:
.
The action of dividing the numerator and denominator of a fraction by the same number is called reducing a fraction:
.
For example: 
; 
.
Work n factors, each of which is equal A, Where A is an arbitrary algebraic expression or real number, and n – natural number, called degreeA :
.
Algebraic expression A called degree basis, number
n – indicator.
For example: 
.
It is believed by definition that for any A, not equal to zero:
And 
.
If 
, That 
.
Properties of degree
1. 
.
2. 
.
3. 
.
4. 
.
5. 
.
If , 
, then the expression n-th degree of which is equal to A, called rootn
th degree ofA
. It is usually denoted 
. Wherein A called radical expression, n called root index.
For example: 
; 
; 
.
Root propertiesnth degree of a
1. 
.
2. 
, 
.
3. 
.
4. 
.
5. 
.
Generalizing the concept of degree and root, we obtain the concept of degree with a rational exponent:
.
In particular, 
.
Actions performed with roots
For example: .
II. Practical material
Examples of completing tasks
Example 1. Find the value of the fraction 
.
Answer: 
.
Example 2. Simplify the expression 
.
Let's transform the expression in the first brackets:
, If 
.
Let's transform the expression in the second brackets:
.
Let's divide the result from the first bracket by the result from the second bracket:
Answer: 
Example 3. Simplify the expression:
.
Example 4. Simplify the expression.
Let's transform the first fraction:
.
Let's transform the second fraction:
.
As a result we get: 
.
Example 5. Simplify the expression 
.
Solution. Let's decide on the following actions:
1) 
;
2) 
;
3) 
;
6) 
;
Answer: 
.
Example 6. Prove the identity 
.
1) 
;
2) 
;
Example 7. Simplify the expression:
.
Solution. Follow these steps:
;
2) 
.
Example 8. Prove the identity 
.
Solution. Follow these steps:
1) 
;
2) 
;
3) 
.
Tasks for independent work
1. Simplify the expression:
A) 
;
b) 
;
2. Factor into:
A) 
;
b) 
;.Document
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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 for given values of x, y, z, you need to perform three steps. 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 have simplified the calculations by replacing the expression xy-xz with the 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 the combinatory property of 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 us represent the second term -(4b-c) in this expression as a product (-1)(4b-c):
a-(4b-c)=a+(-1)(4b-c).
Applying the specified properties of actions, we get:
a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.
I. Expressions in which numbers, arithmetic symbols and parentheses can be used along with letters are called algebraic expressions.
Examples of algebraic expressions:
2m -n; 3 · (2a + b); 0.24x; 0.3a -b · (4a + 2b); a 2 – 2ab;
Since a letter in an algebraic expression can be replaced by some different numbers, the letter is called a variable, and the algebraic expression itself is called an expression with a variable.
II. If in an algebraic expression the letters (variables) are replaced by their values and the specified actions are performed, then the resulting number is called the value of the algebraic expression.
Examples. Find the meaning of the expression:
1) a + 2b -c with a = -2; b = 10; c = -3.5.
2) |x| + |y| -|z| at x = -8; y = -5; z = 6.
Solution.
1) a + 2b -c with a = -2; b = 10; c = -3.5. Instead of variables, let's substitute their values. We get:
— 2+ 2 · 10- (-3,5) = -2 + 20 +3,5 = 18 + 3,5 = 21,5.
2) |x| + |y| -|z| at x = -8; y = -5; z = 6. Substitute the indicated values. Remember that the module negative number is equal to its opposite number, and the modulus of a positive number is equal to this number itself. We get:
|-8| + |-5| -|6| = 8 + 5 -6 = 7.
III. The values of the letter (variable) for which the algebraic expression makes sense are called the permissible values of the letter (variable).
Examples. At what values variable expression doesn't make sense?
Solution. We know that you cannot divide by zero, therefore, each of these expressions will not make sense given the value of the letter (variable) that turns the denominator of the fraction to zero!
In example 1) this value is a = 0. Indeed, if you substitute 0 instead of a, then you will need to divide the number 6 by 0, but this cannot be done. Answer: expression 1) does not make sense when a = 0.
In example 2) the denominator of x is 4 = 0 at x = 4, therefore, this value x = 4 cannot be taken. Answer: expression 2) does not make sense when x = 4.
In example 3) the denominator is x + 2 = 0 when x = -2. Answer: expression 3) does not make sense when x = -2.
In example 4) the denominator is 5 -|x| = 0 for |x| = 5. And since |5| = 5 and |-5| = 5, then you cannot take x = 5 and x = -5. Answer: expression 4) does not make sense at x = -5 and at x = 5.
IV. Two expressions are said to be identically equal if, for any admissible values of the variables, the corresponding values of these expressions are equal.
Example: 5 (a – b) and 5a – 5b are also equal, since the equality 5 (a – b) = 5a – 5b will be true for any values of a and b. The equality 5 (a – b) = 5a – 5b is an identity.
Identity is an equality that is valid for all permissible values of the variables included in it. Examples of identities already known to you are, for example, the properties of addition and multiplication, and the distributive property.
Replacing one expression with another identically equal expression is called an identity transformation or simply a transformation of an expression. Identical transformations of expressions with variables are performed based on the properties of operations on numbers.
Examples.
a) convert the expression to identically equal using the distributive property of multiplication:
1) 10·(1.2x + 2.3y); 2) 1.5·(a -2b + 4c); 3) a·(6m -2n + k).
Solution. Let us recall the distributive property (law) of multiplication:
(a+b)c=ac+bc(distributive law of multiplication relative to addition: in order to multiply the sum of two numbers by a third number, you can multiply each term by this number and add the resulting results).
(a-b) c=a c-b c(distributive law of multiplication relative to subtraction: in order to multiply the difference of two numbers by a third number, you can multiply the minuend and subtract by this number separately and subtract the second from the first result).
1) 10·(1.2x + 2.3y) = 10 · 1.2x + 10 · 2.3y = 12x + 23y.
2) 1.5·(a -2b + 4c) = 1.5a -3b + 6c.
3) a·(6m -2n + k) = 6am -2an +ak.
b) transform the expression into identically equal, using the commutative and associative properties (laws) of addition:
4) x + 4.5 +2x + 6.5; 5) (3a + 2.1) + 7.8; 6) 5.4s -3 -2.5 -2.3s.
Solution. Let's apply the laws (properties) of addition:
a+b=b+a(commutative: rearranging the terms does not change the sum).
(a+b)+c=a+(b+c)(combinative: in order to add a third number to the sum of two terms, you can add the sum of the second and third to the first number).
4) x + 4.5 +2x + 6.5 = (x + 2x) + (4.5 + 6.5) = 3x + 11.
5) (3a + 2.1) + 7.8 = 3a + (2.1 + 7.8) = 3a + 9.9.
6) 6) 5.4s -3 -2.5 -2.3s = (5.4s -2.3s) + (-3 -2.5) = 3.1s -5.5.
V) Convert the expression to identically equal using the commutative and associative properties (laws) of multiplication:
7) 4 · X · (-2,5); 8) -3,5 · 2у · (-1); 9) 3a · (-3) · 2s.
Solution. Let's apply the laws (properties) of multiplication:
a·b=b·a(commutative: rearranging the factors does not change the product).
(a b) c=a (b c)(combinative: 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).