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 us represent all the terms in the form of monomials of the standard form:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16\)
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.
For 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 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
You have to deal with some expressions in algebraic transformations 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 such a 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.
\((a + b)^2 = a^2 + b^2 + 2ab \) - the square of the sum is equal to the sum of the squares and the double product.
\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is equal to the sum of squares without the double 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 in transformations to replace their left parts with right ones and vice versa - right parts with left 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.
Numerical and algebraic expressions. Converting Expressions.
What is an expression in mathematics? Why do we need expression conversions?
The question, as they say, is interesting... The fact is that these concepts are the basis of all mathematics. All mathematics consists of expressions and their transformations. Not very clear? Let me explain.
Let's say you have an evil example in front of you. Very big and very complex. Let's say you're good at math and aren't afraid of anything! Can you give an answer right away?
You will have to decide this example. Consistently, step by step, this example simplify. According to certain rules, of course. Those. do expression conversion. The more successfully you carry out these transformations, the stronger you are in mathematics. If you don't know how to do the right transformations, you won't be able to do them in math. Nothing...
To avoid such an uncomfortable future (or present...), it doesn’t hurt to understand this topic.)
First, let's find out what is an expression in mathematics. What's happened numeric expression and what is algebraic expression.
What is an expression in mathematics?
Expression in mathematics- this is a very broad concept. Almost everything we deal with in mathematics is a set of mathematical expressions. Any examples, formulas, fractions, equations, and so on - it all consists of mathematical expressions.
3+2 is a mathematical expression. s 2 - d 2- this is also a mathematical expression. Both a healthy fraction and even one number are all mathematical expressions. For example, the equation is:
5x + 2 = 12
consists of two mathematical expressions connected by an equal sign. One expression is on the left, the other on the right.
In general, the term " mathematical expression"is used, most often, to avoid mooing. They will ask you what an ordinary fraction is, for example? And how to answer?!
First answer: "This is... mmmmmm... such a thing... in which... Can I write a fraction better? Which one do you want?"
The second answer: “An ordinary fraction is (cheerfully and joyfully!) mathematical expression , which consists of a numerator and a denominator!"
The second option will be somehow more impressive, right?)
This is the purpose of the phrase " mathematical expression "very good. Both correct and solid. But for practical use you need to have a good understanding of specific types of expressions in mathematics .
The specific type is another matter. This It's a completely different matter! Each type of mathematical expression has mine a set of rules and techniques that must be used when making a decision. For working with fractions - one set. For working with trigonometric expressions - the second one. For working with logarithms - the third. And so on. Somewhere these rules coincide, somewhere they differ sharply. But don't be afraid of these scary words. We will master logarithms, trigonometry and other mysterious things in the appropriate sections.
Here we will master (or - repeat, depending on who...) two main types of mathematical expressions. Numerical expressions and algebraic expressions.
Numeric expressions.
What's happened numeric expression? This is a very simple concept. The name itself hints that this is an expression with numbers. Yes, that's how it is. A mathematical expression made up of numbers, brackets and arithmetic symbols is called a numerical expression.
7-3 is a numerical expression.
(8+3.2) 5.4 is also a numerical expression.
And this monster:
also a numerical expression, yes...
An ordinary number, a fraction, any example of calculation without X's and other letters - all these are numerical expressions.
Main sign numerical expressions - in it no letters. None. Only numbers and mathematical symbols (if necessary). It's simple, right?
And what can you do with numerical expressions? Numeric expressions can usually be counted. To do this, it happens that you have to open the brackets, change signs, abbreviate, swap terms - i.e. do expression conversions. But more on that below.
Here we will deal with such a funny case when with a numerical expression you don't need to do anything. Well, nothing at all! This pleasant operation - do nothing)- is executed when the expression doesn't make sense.
When does a numerical expression make no sense?
It’s clear that if we see some kind of abracadabra in front of us, like
then we won’t do anything. Because it’s not clear what to do about it. Some kind of nonsense. Maybe count the number of pluses...
But there are outwardly quite decent expressions. For example this:
(2+3) : (16 - 2 8)
However, this expression also doesn't make sense! For the simple reason that in the second brackets - if you count - you get zero. But you can’t divide by zero! This is a forbidden operation in mathematics. Therefore, there is no need to do anything with this expression either. For any task with such an expression, the answer will always be the same: "The expression has no meaning!"
To give such an answer, of course, I had to calculate what would be in brackets. And sometimes there’s a lot of stuff in parentheses... Well, there’s nothing you can do about it.
There are not so many forbidden operations in mathematics. There is only one in this topic. Division by zero. Additional restrictions arising in roots and logarithms are discussed in the corresponding topics.
So, an idea of what it is numeric expression- received. Concept the numeric expression doesn't make sense- realized. Let's move on.
Algebraic expressions.
If letters appear in a numerical expression, this expression becomes... The expression becomes... Yes! It becomes algebraic expression. For example:
5a 2; 3x-2y; 3(z-2); 3.4m/n; x 2 +4x-4; (a+b) 2; ...
Such expressions are also called literal expressions. Or expressions with variables. It's practically the same thing. Expression 5a +c, for example, both literal and algebraic, and an expression with variables.
Concept algebraic expression - broader than numeric. It includes and all numerical expressions. Those. a numerical expression is also an algebraic expression, only without letters. Every herring is a fish, but not every fish is a herring...)
Why alphabetic- It's clear. Well, since there are letters... Phrase expression with variables It’s also not very puzzling. If you understand that numbers are hidden under the letters. All sorts of numbers can be hidden under letters... And 5, and -18, and anything else. That is, a letter can be replace for different numbers. That's why the letters are called variables.
In expression y+5, For example, at- variable value. Or they just say " variable", without the word "magnitude". Unlike five, which is a constant value. Or simply - constant.
Term algebraic expression means that to work with this expression you need to use laws and rules algebra. If arithmetic works with specific numbers, then algebra- with all the numbers at once. A simple example for clarification.
In arithmetic we can write that
But if we write such an equality through algebraic expressions:
a + b = b + a
we'll decide right away All questions. For all numbers in one fell swoop. For everything infinite. Because under the letters A And b implied All numbers. And not only numbers, but even other mathematical expressions. This is how algebra works.
When does an algebraic expression not make sense?
Everything about the numerical expression is clear. You can't divide by zero there. And with letters, is it possible to find out what we are dividing by?!
Let's take for example this expression with variables:
2: (A - 5)
Does it make sense? Who knows? A- any number...
Any, any... But there is one meaning A, for which this expression exactly doesn't make sense! And what is this number? Yes! This is 5! If the variable A replace (they say “substitute”) with the number 5, in brackets you get zero. Which cannot be divided. So it turns out that our expression doesn't make sense, If a = 5. But for other values A does it make sense? Can you substitute other numbers?
Certainly. In such cases they simply say that the expression
2: (A - 5)
makes sense for any values A, except a = 5 .
The whole set of numbers that Can substituting into a given expression is called range of acceptable values this expression.
As you can see, there is nothing tricky. Let's look at the expression with variables and figure out: at what value of the variable is the forbidden operation (division by zero) obtained?
And then be sure to look at the task question. What are they asking?
doesn't make sense, our forbidden meaning will be the answer.
If you ask at what value of a variable the expression makes sense(feel the difference!), the answer will be all other numbers except for the forbidden.
Why do we need the meaning of the expression? He is there, he is not... What's the difference?! The point is that this concept becomes very important in high school. Extremely important! This is the basis for such solid concepts as the domain of acceptable values or the domain of a function. Without this, you will not be able to solve serious equations or inequalities at all. Like this.
Converting Expressions. Identity transformations.
We were introduced to numerical and algebraic expressions. We understood what the phrase “the expression has no meaning” means. Now we need to figure out what it is transformation of expressions. The answer is simple, to the point of disgrace.) This is any action with an expression. That's all. You have been doing these transformations since first grade.
Let's take the cool numerical expression 3+5. How can it be converted? Yes, very simple! Calculate:
This calculation will be the transformation of the expression. You can write the same expression differently:
Here we didn’t count anything at all. Just wrote down the expression in a different form. This will also be a transformation of the expression. You can write it like this:
And this too is a transformation of an expression. You can make as many such transformations as you want.
Any action on expression any writing it in another form is called transforming the expression. And that's all. It's very simple. But there is one thing here very important rule. So important that it can safely be called main rule all mathematics. Breaking this rule inevitably leads to errors. Are we getting into it?)
Let's say we transformed our expression haphazardly, like this:
Conversion? Certainly. We wrote the expression in a different form, what’s wrong here?
It's not like that.) The point is that transformations "at random" are not interested in mathematics at all.) All mathematics is built on transformations in which the appearance changes, but the essence of the expression does not change. Three plus five can be written in any form, but it must be eight.
Transformations, expressions that do not change the essence are called identical.
Exactly identity transformations and allow us, step by step, to transform a complex example into a simple expression, while maintaining the essence of the example. If we make a mistake in the chain of transformations, we make a NOT identical transformation, then we will decide another example. With other answers that are not related to the correct ones.)
This is the main rule for solving any tasks: maintaining the identity of transformations.
I gave an example with the numerical expression 3+5 for clarity. In algebraic expressions, identity transformations are given by formulas and rules. Let's say in algebra there is a formula:
a(b+c) = ab + ac
This means that in any example we can instead of the expression a(b+c) feel free to write an expression ab + ac. And vice versa. This identical transformation. Mathematics gives us a choice between these two expressions. And which one to write depends on the specific example.
Another example. One of the most important and necessary transformations is the basic property of a fraction. You can look at the link for more details, but here I’ll just remind you of the rule: If the numerator and denominator of a fraction are multiplied (divided) by the same number, or an expression that is not equal to zero, the fraction will not change. Here is an example of identity transformations using this property:
As you probably guessed, this chain can be continued indefinitely...) A very important property. It is this that allows you to turn all sorts of example monsters into white and fluffy.)
There are many formulas defining identical transformations. But the most important ones are quite a reasonable number. One of the basic transformations is factorization. It is used in all mathematics - from elementary to advanced. Let's start with him. In the next lesson.)
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
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.
References.
- 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. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
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: in order 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 a numerical expression of the form a-b to be considered the sum of numbers a and -b, a numerical expression of the form a+b-c-d to 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 identical 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 obtain:
a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.
