Definition 3.3. Monomial called an expression that is the product of numbers, variables and powers with natural indicator.
For example, each of the expressions, 
,
is a monomial.
They say that the monomial has standard view , if it contains only one numerical factor in the first place, and each product of identical variables in it is represented by a degree. The numerical factor of a monomial written in standard form is called coefficient of the monomial . By the power of the monomial is called the sum of the exponents of all its variables.
Definition 3.4. Polynomial called the sum of monomials. The monomials from which a polynomial is composed are calledmembers of the polynomial .
Similar terms - monomials in a polynomial - are called similar terms of the polynomial .
Definition 3.5. Polynomial of standard form called a polynomial in which all terms are written in standard form and similar terms are given.Degree of a polynomial of standard form is called the greatest of the powers of the monomials included in it.
For example, is a polynomial of standard form of the fourth degree.
Actions on monomials and polynomials
The sum and difference of polynomials can be converted into a polynomial of standard form. When adding two polynomials, all their terms are written down and similar terms are given. When subtracting, the signs of all terms of the polynomial being subtracted are reversed.
For example:
The terms of a polynomial can be divided into groups and enclosed in parentheses. Since this is an identical transformation inverse to the opening of parentheses, the following is established bracketing rule: if a plus sign is placed before the brackets, then all terms enclosed in brackets are written with their signs; If a minus sign is placed before the brackets, then all terms enclosed in brackets are written with opposite signs.
For example,
Rule for multiplying a polynomial by a polynomial: To multiply a polynomial by a polynomial, it is enough to multiply each term of one polynomial by each term of another polynomial and add the resulting products.
For example,
Definition 3.6. Polynomial in one variable 
degrees 
called an expression of the form
Where 
- any numbers that are called polynomial coefficients
, and 
,
– non-negative integer.
If 
, then the coefficient 
called leading coefficient of the polynomial
, monomial 
- his senior member
, coefficient 
–
free member
.
If instead of a variable 
to a polynomial 
substitute real number 
, then the result will be a real number 
which is called the value of the polynomial
at 
.
Definition 3.7.
Number 
calledroot of the polynomial
, If
.
Consider dividing a polynomial by a polynomial, where 
And 
- integers. Division is possible if the degree of the polynomial dividend is 
Not less degree divisor polynomial 
, that is 
.
Divide a polynomial 
to a polynomial 
,
, means finding two such polynomials 
And 
, to
In this case, the polynomial 
degrees 
called polynomial-quotient
,
–
the remainder
,
.
Remark 3.2.
If the divisor
–is not a zero polynomial, then division 
on 
,
, is always feasible, and the quotient and remainder are uniquely determined.
Remark 3.3.
In case
in front of everyone 
, that is
they say that it is a polynomial
completely divided(or shares)to a polynomial
.
The division of polynomials is carried out similarly to the division of multi-digit numbers: first, the leading term of the dividend polynomial is divided by the leading term of the divisor polynomial, then the quotient from the division of these terms, which will be the leading term of the quotient polynomial, is multiplied by the divisor polynomial and the resulting product is subtracted from the dividend polynomial . As a result, a polynomial is obtained - the first remainder, which is divided by the divisor polynomial in a similar way and the second term of the quotient polynomial is found. This process is continued until a zero remainder is obtained or the degree of the remainder polynomial is less than the degree of the divisor polynomial.
When dividing a polynomial by a binomial, you can use Horner's scheme.
Horner scheme
Suppose we want to divide a polynomial
by binomial 
. Let us denote the quotient of division as a polynomial
and the remainder - 
. Meaning 
, polynomial coefficients 
,
and the remainder 
Let's write it in the following form:
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In this scheme, each of the coefficients
,
,
,
…,
obtained from previous date bottom line multiplied by number 
and adding to the resulting result the corresponding number in the top line above the desired coefficient. If any degree 
is absent in the polynomial, then the corresponding coefficient equal to zero. Having determined the coefficients according to the given scheme, we write the quotient
and the result of division if 
,
or ,
If 
,
Theorem 3.1.
In order for an irreducible fraction 
(
,
)was the root of the polynomial 
with integer coefficients, it is necessary that the number 
was a divisor of the free term 
, and the number 
- divisor of the leading coefficient 
.
Theorem 3.2.
(Bezout's theorem
)
Remainder 
from dividing a polynomial 
by binomial 
equal to the value of the polynomial 
at 
, that is 
.
When dividing a polynomial 
by binomial 
we have equality
This is true, in particular, when 
, that is 
.
Example 3.2. Divide by 
.
Solution. Let's apply Horner's scheme:
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Hence,
Example 3.3. Divide by 
.
Solution. Let's apply Horner's scheme:
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Hence,
,
Example 3.4. Divide by 
.
Solution.
As a result we get
Example 3.5. Divide 
on 
.
Solution. Let's divide the polynomials by column:
|
|
Then we get
.
Sometimes it is useful to represent a polynomial as an equal product of two or more polynomials. Such an identity transformation is called factoring a polynomial . Let us consider the main methods of such decomposition.
Taking the common factor out of brackets. In order to factor a polynomial by taking the common factor out of brackets, you must:
1) find the common factor. To do this, if all the coefficients of the polynomial are integers, the largest modulo common divisor of all coefficients of the polynomial is considered as the coefficient of the common factor, and each variable included in all terms of the polynomial is taken with the largest exponent it has in this polynomial;
2) find the quotient of division given polynomial by a common factor;
3) write down the product of the general factor and the resulting quotient.
Grouping of members. When factoring a polynomial using the grouping method, its terms are divided into two or more groups so that each of them can be converted into a product, and the resulting products would have a common factor. After this, the method of bracketing the common factor of the newly transformed terms is used.
Application of abbreviated multiplication formulas. In cases where the polynomial to be expanded into factors, has the form of the right side of any abbreviated multiplication formula; its factorization is achieved by using the corresponding formula written in a different order.
Let 
, then the following are true abbreviated multiplication formulas:
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For |
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If |
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Newton binomial: Where |
|
:
odd ( 
):
– number of combinations of 
By 
.Introduction of new auxiliary members. This method consists in replacing a polynomial with another polynomial that is identically equal to it, but containing a different number of terms, by introducing two opposite terms or replacing any term with an identically equal sum of similar monomials. The replacement is made in such a way that the method of grouping terms can be applied to the resulting polynomial.
Example 3.6..
Solution. All terms of a polynomial contain a common factor 
. Hence,.
Answer: .
Example 3.7.
Solution. We group separately the terms containing the coefficient 
, and terms containing 
. Bracketing common factors groups, we get:
.
Answer:
.
Example 3.8. Factor a polynomial 
.
Solution. Using the appropriate abbreviated multiplication formula, we get:
Answer: .
Example 3.9. Factor a polynomial 
.
Solution. Using the grouping method and the corresponding abbreviated multiplication formula, we obtain:
.
Answer: .
Example 3.10. Factor a polynomial 
.
Solution. We will replace 
on 
, group the terms, apply the abbreviated multiplication formulas:
.
Answer:
.
Example 3.11. Factor a polynomial
Solution. Because , 
,
, That
Lesson on the topic: "Standard form of a monomial. Definition. Examples"
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Monomial. Definition
Monomial- This mathematical expression, which is the product prime factor and one or more variables.Monomials include all numbers, variables, their powers with a natural exponent:
42; 3; 0; 6 2 ; 2 3 ; b 3 ; ax 4 ; 4x 3 ; 5a 2 ; 12xyz 3 .
Quite often it is difficult to determine whether a given mathematical expression refers to a monomial or not. For example, $\frac(4a^3)(5)$. Is this a monomial or not? To answer this question we need to simplify the expression, i.e. present in the form: $\frac(4)(5)*a^3$.
We can say for sure that this expression- monomial
Standard form of monomial
When calculating, it is desirable to reduce the monomial to standard view. This is the most concise and understandable recording of a monomial.The procedure for reducing a monomial to standard form is as follows:
1. Multiply the coefficients of the monomial (or numerical factors) and place the resulting result in first place.
2. Select all powers with the same letter base and multiply them.
3. Repeat point 2 for all variables.
Examples.
I. Reduce the given monomial $3x^2zy^3*5y^2z^4$ to standard form.
Solution.
1. Multiply the coefficients of the monomial $15x^2y^3z * y^2z^4$.
2. Now let's give similar terms$15х^2y^5z^5$.
II. Reduce the given monomial $5a^2b^3 * \frac(2)(7)a^3b^2c$ to standard form.
Solution.
1. Multiply the coefficients of the monomial $\frac(10)(7)a^2b^3*a^3b^2c$.
2. Now we present similar terms $\frac(10)(7)a^5b^5c$.
Power of a monomial
For a monomial there is the concept of its degree. Let's figure out what it is.
Definition.
Power of a monomial standard form is the sum of exponents of all variables included in its record; if there are no variables in the notation of a monomial and it is different from zero, then its degree is considered equal to zero; the number zero is considered a monomial whose degree is undefined.
Determining the degree of a monomial allows you to give examples. The degree of the monomial a is equal to one, since a is a 1. The power of the monomial 5 is zero, since it is non-zero and its notation does not contain variables. And the product 7·a 2 ·x·y 3 ·a 2 is a monomial of the eighth degree, since the sum of the exponents of all variables a, x and y is equal to 2+1+3+2=8.
By the way, the degree of a monomial not written in standard form is equal to the degree of the corresponding monomial of standard form. To illustrate this, let us calculate the degree of the monomial 3 x 2 y 3 x (−2) x 5 y. This monomial in standard form has the form −6·x 8 ·y 4, its degree is 8+4=12. Thus, the degree of the original monomial is 12.
Monomial coefficient
A monomial in standard form, which has at least one variable in its notation, is a product with a single numerical factor - a numerical coefficient. This coefficient is called the monomial coefficient. Let us formulate the above arguments in the form of a definition.
Definition.
Monomial coefficient is the numerical factor of a monomial written in standard form.
Now we can give examples of coefficients of various monomials. The number 5 is the coefficient of the monomial 5·a 3 by definition, similarly the monomial (−2,3)·x·y·z has a coefficient of −2,3.
The coefficients of the monomials, equal to 1 and −1, deserve special attention. The point here is that they are usually not explicitly present in the recording. It is believed that the coefficient of monomials of the standard form, which do not have a numerical factor in their notation, equal to one. For example, monomials a, x·z 3, a·t·x, etc. have a coefficient of 1, since a can be considered as 1·a, x·z 3 - as 1·x·z 3, etc.
Similarly, the coefficient of monomials, the entries of which in standard form do not have a numerical factor and begin with a minus sign, is considered to be minus one. For example, monomials −x, −x 3 y z 3, etc. have a coefficient −1, since −x=(−1) x, −x 3 y z 3 =(−1) x 3 y z 3 and so on.
By the way, the concept of the coefficient of a monomial is often referred to as monomials of the standard form, which are numbers without letter factors. The coefficients of such monomials-numbers are considered to be these numbers. So, for example, the coefficient of the monomial 7 is considered equal to 7.
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.
- 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.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
We noted that any monomial can be bring to standard form. In this article we will understand what is called bringing a monomial to standard form, what actions allow this process to be carried out, and consider solutions to examples with detailed explanations.
Page navigation.
What does it mean to reduce a monomial to standard form?
It is convenient to work with monomials when they are written in standard form. However, quite often monomials are specified in a form different from the standard one. In these cases, you can always go from the original monomial to a monomial of the standard form by performing identity transformations. The process of carrying out such transformations is called reducing a monomial to a standard form.
Let us summarize the above arguments. Reduce the monomial to standard form- this means doing the following with him identity transformations so that it takes the standard form.
How to bring a monomial to standard form?
It's time to figure out how to reduce monomials to standard form.
As is known from the definition, monomials of non-standard form are products of numbers, variables and their powers, and possibly repeating ones. And a monomial of the standard form can contain in its notation only one number and non-repeating variables or their powers. Now it remains to understand how to bring products of the first type to the type of the second?
To do this you need to use the following the rule for reducing a monomial to standard form consisting of two steps:
- First, a grouping of numerical factors is performed, as well as identical variables and their powers;
- Secondly, the product of the numbers is calculated and applied.
As a result of applying the stated rule, any monomial will be reduced to a standard form.
Examples, solutions
All that remains is to learn how to apply the rule from previous paragraph when solving examples.
Example.
Reduce the monomial 3 x 2 x 2 to standard form.
Solution.
Let's group numerical factors and factors with a variable x. After grouping, the original monomial will take the form (3·2)·(x·x 2) . The product of the numbers in the first brackets is equal to 6, and the rule for multiplying powers with on the same grounds allows the expression in the second brackets to be represented as x 1 +2=x 3. As a result, we obtain a polynomial of the standard form 6 x 3.
Here is a short summary of the solution: 3 x 2 x 2 =(3 2) (x x 2)=6 x 3.
Answer:
3 x 2 x 2 =6 x 3.
So, to bring a monomial to a standard form, you need to be able to group factors, multiply numbers, and work with powers.
To consolidate the material, let's solve one more example.
Example.
Present the monomial in standard form and indicate its coefficient.
Solution.
The original monomial has a single numerical factor in its notation −1, let's move it to the beginning. After this, we will separately group the factors with the variable a, separately with the variable b, and there is nothing to group the variable m with, we will leave it as is, we have 
. After performing operations with powers in brackets, the monomial will take the standard form we need, from which we can see the coefficient of the monomial equal to −1. Minus one can be replaced with a minus sign: .