What does it mean to submit a polynomial in standard form. Polynomials

The polynomial call the amount of single-wing. If all members of the polynomial are recorded in the standard form (see clause 51) and bringing such members, then the polynomial is standard.

A whole expression can be converted to a polynomial of a standard species - this is the purpose of transformations (simplifications) of integer expressions.

Consider examples in which an integer expression should be brought to the standard type of polynomial.

Decision. First we give the standard member of the polynomial. We get after bringing such members to get a polynomial of a standard species

Decision. If there is a sign "plus, then the brackets can be omitted by saving the signs of all the terms enrolled in brackets. Using this rule of disclosure of the brackets, we get:

Decision. If Ziac "minus" stands in front of the brackets, then brackets can be omitted by changing the signs of all the terms "prisoners in brackets. Taking advantage of this rules of the packering brackets, we get:

Decision. The work is unoblant and polynomial according to the distributional law equal to the amount of works of this single and each member of the polynomial. Receive

Decision. Have

It remains to bring similar members (they are underlined). We get:

53. Formulas of abbreviated multiplication.

In some cases, the accuracy of a whole expression to the standard type of polynomial is carried out using identities:

These identities call the formulas of abbreviated multiplication,

Consider examples in which you need to convert a specified expression in the Mooroche of the standard view.

Example 1..

Decision. Using formula (1), we get:

Example 2..

Decision.

Example 3..

Decision. Using formula (3), we get:

Example 4.

Decision. Using formula (4), we get:

54. Decomposition of polynomials on multipliers.

Sometimes it is possible to transform a polynomial into the work of several factors - polynomials or related. This identical transformation is called the decomposition of the polynomial to multipliers. In this case, they say that the polynomial is divided into each of these factors.

Consider some ways of decomposing polynomials for multipliers,

1) Making a common factor behind the bracket. This transformation is a direct consequence of the distribution law (for clarity it is necessary only to rewrite this law "Right to the left"):

Example 1. Defix many polynomials

Decision. .

Usually, when issuing a common factor for parentheses, each variable that is included in all members of the polynomial is carried out with the smallest indicator that it has in this polynomial. If all the polynomial coefficients are integers, then the total divisor of all polynomial coefficients is taken as the total multi-module coefficient.

2) the use of formulas of abbreviated multiplication. Formulas (1) - (7) from clause 53, being read "right to left, in many cases are beneficial to decompose polynomials to multipliers.

Example 2. Disintegrate on multipliers.

Decision. We have. Applying formula (1) (the difference of squares), we get. Apply

now formulas (4) and (5) (the amount of cubes, the difference of cubes), we get:

Example 3..

Decision. First I will bring a general multiplier for the bracket. To do this, we find the greatest overall divisor of the coefficients 4, 16, 16 and the smallest indicators of the degrees with which the variables A and B are included in the components of this polynomial universal. We get:

3) Grouping method. It is based on the fact that the transitional and combination laws of addition make it possible to group members of the polynomial in various ways. Sometimes such a grouping is possible that after making the brackets of general multipliers, the same polynomial remains in brackets in parentheses, which in turn as a common multiplier can be rendered per brackets. Consider examples of decomposition of polynomials on multipliers.

Example 4..

Decision. Produce a grouping as follows:

In the first group, I will carry out a general factor in the second group in the second - a general factor 5. We will now get a polynomial as a general factor I will bring it for a bracket: Thus, we get:

Example 5.

Decision. .

Example 6.

Decision. Here, no grouping will lead to the appearance in all groups of the same polynomial. In such cases, it is sometimes useful to submit any member of the polynomial in the form of a certain amount, after which try again to apply the grouping method. In our example, it is advisable to submit in the form of the amount we get

Example 7.

Decision. I will add and take away one-time

55. Mounted from one variable.

Polynomial, where a, b is the number of variable, called the first degree polynomial; polynomial where a, b, C - number variable, is called a polynomial of a second degree or square three decrease; The polynomial where a, b, c, d is the variable is called a polynomial of a third degree.

In general, if about, variable, then polynomial

called Lsmochlenol degree (relative to x); , M-members of the polynomial, coefficients, senior member of the polynomial, and the coefficient with the senior member, a free member of the polynomial. Typically, the polynomial is recorded on decreasing degrees of the variable, i.e. the degrees of the variable gradually decrease, in particular, in the first place there is a senior dick, on the latter - a free member. The degree of polynomial is the degree of senior member.

For example, a polynomial of the fifth degree in which the senior dick, 1 is a free member of the polynomial.

The root of the polynomial calls such a value at which the polynomial adds to zero. For example, the number 2 is the root of the polynomial since

After studying homorals, we turn to polynomials. This article will tell about all the necessary information necessary to perform actions on them. We define a polynomial with the accompanying definitions of a member of the polynomial, that is, free and similar, consider the polynomial of the standard species, we introduce a degree and learn how to find it, we will work with its coefficients.

The polynomials and its members - definitions and examples

The definition of the polynomial was given in 7 class after studying homorals. Consider its full definition.

Definition 1.

Polynomial The amount of single-wing is considered, and it is unrigous himself - this is a special case of a polynomial.

It follows from the definition that examples of polynomials may be different: 5 , 0 , − 1 , X., 5 · A · B 3, x 2 · 0, 6 · x · (- 2) · y 12, - 2 13 · x · y 2 · 3 2 3 · x · x 3 · y · z and so on. From the definition we have that 1 + X., a 2 + b 2 and the expression x 2 - 2 · x · y + 2 5 · x 2 + y 2 + 5, 2 · y · x are polynomials.

Consider still definitions.

Definition 2.

Members of the polynomialit is called its components shared.

Consider such an example, where we have a polynomial 3 · x 4 - 2 · x · y + 3 - y 3, consisting of 4 members: 3 · x 4, - 2 · x · y, 3 and - Y 3.. Such a single one can be considered a polynomial, which consists of one member.

Definition 3.

Polynomials that are in their composition 2, 3 three declections have a respective name - binomial and trinomial.

Hence it follows that the expression of the form X + Y.- It is twisted, and the expression 2 · x 3 · q - q · x · x + 7 · b is threehow.

By school Program Worked with linear biccourse of the form A · X + B, where a and b are some numbers, and x - variable. Consider the examples of linear two-dimensions of the form: x + 1, x · 7, 2 - 4 with examples of square three-strokes x 2 + 3 · x - 5 and 2 5 · x 2 - 3 x + 11.

For conversion and solutions it is necessary to find and bring similar terms. For example, a polynomial of the form 1 + 5 · X - 3 + Y + 2 · X has the similar terms of 1 and - 3, 5 x and 2 x. They are divided into a special group called similar members of the polynomial.

Definition 4.

Similar members of the polynomial- These are similar components that are in the polynomial.

In the example above, we have that 1 and - 3, 5 x and 2 x are similar members of the polynomial or similar terms. In order to simplify the expression, it is used to find and bring similar terms.

Polynomial standard view

All single-sided and polynomials have their own definite names.

Definition 5.

Polynomial standard viewthey call a polynomial, in which each member part in it has a single standard species and does not contain such members.

It can be seen from the definition that it is possible to bring the polynomials of the standard species, for example, 3 · x 2 - x · y + 1 and __formula__, and the recording is standard. Expressions 5 + 3 · x 2 - x 2 + 2 · x · z and 5 + 3 · x 2 - x 2 + 2 · x · z polynomials of the standard species is not, since the first of them has similar terms in the form of 3 · x 2 I. - X 2., and the second contains a single form x · y 3 · x · z 2, differing from the standard polynomial.

If the circumstances require, sometimes the polynomial is reduced to the standard form. The concept of a free member of the polynomial is also considered a polynomial.

Definition 6.

Free member of the polynomialit is a polynomial of a standard species that does not have an alphabetic part.

In other words, when the recording of the polynomial in the standard form has a number, it is called a free member. Then the number 5 is a free member of the polynomial x 2 · Z + 5, and the polynomial 7 · a + 4 · a · b + b 3 does not have a free member.

The degree of polynomial - how to find it?

The determination of the degree of the polynomial is based on the definition of the polynomial of the standard type and on the degrees of single-wing, which are its constituent.

Definition 7.

The degree of polynomial of the standard typecall the greatest of the degrees that are included in its recording.

Consider on the example. The degree of polynomial 5 · x 3 - 4 is 3, because they are notched, which are included in its composition, have degrees 3 and 0, and more of them 3, respectively. The determination of the degree of polynomial 4 · x 2 · y 3 - 5 · x 4 · y + 6 · x is equal to the largest of the numbers, that is, 2 + 3 \u003d 5, 4 + 1 \u003d 5 and 1, it means 5.

It should be found in how the degree is located.

Definition 8.

The degree of polynomial of an arbitrary number - This is the degree of corresponding polynomial in standard form.

When the polynomial is not recorded not in the standard form, but it is necessary to find its degree, it is necessary to bring to the standard one, after which find a desired degree.

Example 1.

Find a polynomial 3 · A 12 - 2 · A · B · C · A · C · B + Y 2 · Z 2 - 2 · A 12 - A 12.

Decision

First, submit a polynomial in standard form. We obtain the expression of the form:

3 · A 12 - 2 · A · B · C · A · C · B + Y 2 · Z 2 - 2 · A 12 - A 12 \u003d (3 · A 12 - 2 · A 12 - A 12) - 2 · (A) · (b · b) · (C · C) + y 2 · z 2 \u003d \u003d - 2 · a 2 · b 2 · C 2 + y 2 · z 2

When obtaining a polynomial of a standard species, we obtain that two of them are distinctly distinguished - 2 · a 2 · b 2 · C 2 and Y 2 · Z 2. To find degrees, we consider and obtain that 2 + 2 + 2 \u003d 6 and 2 + 2 \u003d 4. It can be seen that the largest of them is equal to 6. It follows from the definition that it is 6 is the degree of polynomial - 2 · a 2 · b 2 · C 2 + Y 2 · Z 2, therefore, the initial value.

Answer: 6 .

The coefficients of members of the polynomial

Definition 9.

When all members of the polynomial are classified as standard, then in this case they are called the coefficients of members of the polynomial.In other words, they can be called polynomial coefficients.

When considering the example, it can be seen that the polynomial of the form 2 · x - 0, 5 · x · y + 3 · x + 7 has 4 polynomials in its composition: 2 · x, - 0, 5 · x · y, 3 · x and 7 With the corresponding coefficients 2, - 0, 5, 3 and 7. So, 2, - 0, 5, 3 and 7 are considered to be coefficients of members of a given polynomial of the form 2 · x - 0, 5 · x · y + 3 · x + 7. When converting it is important to pay attention to the coefficients facing variables.

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We said that they take place both polynomials of a standard species and not standard. There we noted that any polynomial lead to standard. In this article, we will definitely find out what sense this phrase carries. Further list the steps that allow you to transform any polynomial to the standard view. Finally, consider solutions of characteristic examples. Decisions will be described in very detailed to deal with all the nuances that occur when the polynomials are brought to the standard form.

Navigating page.

What does it mean to bring a polynomial to the standard mind?

First, it is necessary to clearly understand what is understood under the presentation of the polynomial to the standard form. Tell me.

Numerous, like any other expressions, can be subjected to identical transformations. As a result of the implementation of such transformations, expressions are obtained, identically equal to the initial expression. So the performance of certain transformations with polynomials is not standard to go to the identical to the polynomials, but recorded already in standard form. Such a transition and call the polynomial to the standard form.

So, lead polynomial to standard - This means replacing the original polynomial identically equal to it by a polynomial of a standard view derived from the initial path of identical transformations.

How to bring a polynomial to the standard form?

Let's think about which transformations will help us to bring polynomial to the standard form. We will be repelled from the definition of the polynomial of the standard species.

By definition, each member of the standard species is a single standard form, and the polynomial of the standard species does not contain such members. In turn, the polynomials recorded in the form other than the standard may consist of single-panels in non-standard form and may contain similar members. From here logically follows the following rule explaining how to bring a polynomial to the standard form:

first, it is necessary to bring to the standard form of universal, of which the original polynomial consists,
after that, perform the creation of such members.

As a result, a polynomial of a standard species will be obtained, since all its members will be recorded in a standard form, and it will not contain similar members.

Examples, solutions

Consider examples of bringing polynomials to the standard form. When solving, we will perform steps dictated by the rule from the previous paragraph.

Here we note that sometimes all members of the polynomial are immediately recorded in a standard form, in this case it is enough just to bring similar members. Sometimes after bringing members of the polynomial to the standard form, there are no such members, therefore, the stage of bringing such members in this case is omitted. In general, you have to do both.

Example.

Imagine polynomials in standard form: 5 · x 2 · y + 2 · y 3 -x · y + 1, 0.8 + 2 · a 3 · 0,6-b · A · B 4 · b 5 and.

Decision.

All members of the polynomial 5 · x 2 · y + 2 · y 3 -x · y + 1 are recorded in standard form, it does not have such members, therefore, this polynomial is already presented in standard form.

Go to the next polynomial 0.8 + 2 · a 3 · 0,6-b · A · B 4 · b 5. Its species is not standard, as evidenced by members 2 · a 3 · 0.6 and -b · a · b 4 · b 5 is not standard. Imagine it in standard form.

At the first stage of bringing the initial polynomial to the standard form, we need to submit all its members in the standard form. Therefore, we present to the standard form. 2 · a 3 · 0.6, we have 2 · a 3 · 0.6 \u003d 1.2 · a 3, after which - unrochene -b · a · b 4 · b 5, we have -B · a · b 4 · b 5 \u003d -a · b 1 + 4 + 5 \u003d -a · b 10. In this way, . In the resulting polynomial, all members are recorded in standard form, moreover, it is obvious that there are no similar members in it. Therefore, this completed bringing the initial polynomial to the standard form.

It remains to present in the standard form the last of the specified polynomials. After bringing all his members to the standard form, he will be recorded as . It has similar members, so you need to carry out similar members:

So the initial polynomial accepted the standard form -x · y + 1.

Answer:

5 · x 2 · y + 2 · y 3 -x · y + 1 - already in standard form, 0.8 + 2 · a 3 · 0,6-b · a · b 4 · b 5 \u003d 0.8 + 1,2 · a 3 -a · b 10, .

Often bringing the polynomial to the standard form is only an intermediate step in response to the task assigned question. For example, the degree of polynomial implies its preview in standard form.

Example.

Give a polynomial To the standard species, specify its degree and place members on decreasing degrees of the variable.

Decision.

First give all members of the polynomial to the standard form: .

Now we give such members:

So we led the original polynomial to the standard form, it allows us to determine the degree of polynomial, which is equal to the greatest degree of universal in it. Obviously, it is equal to 5.

It remains to position the members of the polynomial at decreasing degrees of variables. To do this, it is only necessary to rearrange member members in the obtained polynomial of the standard species, given the requirement. The greatest degree of member Z 5, the degree of members, -0.5 · z 2 and 11 are equal, respectively, 3, 2 and 0. Therefore, the polynomial with the member variable located on decreasing degrees will be .

Answer:

The degree of polynomial is 5, and after the location of its members on decreasing degrees of the variable, it takes .

Bibliography.

Algebra: studies. for 7 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorov]; Ed. S. A. Telikovsky. - 17th ed. - M.: Enlightenment, 2008. - 240 s. : IL. - ISBN 978-5-09-019315-3.
Mordkovich A. G. Algebra. 7th grade. In 2 tsp. 1. Tutorial for students general educational institutions / A. Mordkovich. - 17th ed., Extras - M.: Mnemozina, 2013. - 175 p.: Il. ISBN 978-5-346-02432-3.
Algebra And beginning mathematical analysis. Grade 10: studies. For general education. Institutions: Basic and Profile. Levels / [Y. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; Ed. A. B. Zhizchenko. - 3rd ed. - M.: Enlightenment, 2010.- 368 p. : IL. - ISBN 978-5-09-022771-1.
Gusev V. A., Mordkovich A. G. Mathematics (benefit for applicants in technical schools): studies. benefit. - m.; Higher. Shk., 1984.-351 p., Il.

At this lesson, we will recall the basic definitions of this topic and consider some typical tasks, namely the clarification of the polynomial to the standard form and calculate the numerical value at the specified values \u200b\u200bof the variables. We will solve several examples that will apply to the standard form to solve various types of tasks.

Subject:Polynomials. Arithmetic operations over single-wing

Lesson:Bringing a polynomial to the standard form. Typical tasks

Recall the basic definition: polynomial is the amount of single-wing. Each single-wing, which is part of the polynomial as a component is called his member. For example:

Binomial;

Polynomial;

Binomial;

Since the polynomial consists of single-wing, the first action with a polynomial should be from here - you need to bring everything to the standard form. Recall that for this you need to multiply all the numerical multipliers - to obtain a numerical coefficient, and multiply the appropriate degrees - to obtain an alphabet part. In addition, we will pay attention to the theorem on the work of degrees: when multiplying degrees, their indicators are folded.

Consider an important operation - bringing a polynomial to the standard form. Example:

Comment: To bring a polynomial to the standard form, you need to lead to a standard form. All are unarranged, which are included in its composition, after that, if there are similar unripes - and these are unknown with the same alphabone part - perform actions with them.

So, we looked at the first type task - bringing the polynomial to the standard form.

The following typical task is the calculation of the specific value of the polynomial at a given numerical values \u200b\u200bof the variables included in it. We will continue to consider the previous example and set the values \u200b\u200bof the variables:

Comment: Recall that the unit in any naturally is equal to one, and zero to any natural degree is zero, in addition, we recall that when you multiply any number to zero we get zero.

Consider a number of examples on typical operations of bringing a polynomial to the standard form and the calculation of its value:

Example 1 - lead to standard form:

Comment: First action - we give shake to the standard form, you need to bring the first, second and sixth; The second action - we give such members, that is, we perform a given arithmetic actions on them: the first we fold with the fifth, the second one with the third, the rest rewrite without changes, since they do not have the like.

Example 2 - Calculate the value of the polynomial from Example 1 at the specified values \u200b\u200bof the variables:

Comment: When calculating, it should be remembered that the unit in any natural extent is the unit, with the difficulty of calculations of the degree detection, you can use the degree table.

Example 3 - Instead of an asterisk, put such a single thing so that the result contained the variable:

Comment: regardless of the task, the first action is always the same - to bring polynomial to the standard form. In our example, this action is reduced to bringing similar members. After that, it should be carefully reading the condition and think about how we can get rid of union. It is obvious that for this you need to add the same one to it, but with the opposite sign -. Next, we replace the asterisk with this onemalary and make sure the correctness of our solution.

Definition 3.3. Solid they call an expression, which is a product of numbers, variables and degrees with a natural indicator.

For example, each of the expressions,
,
it is single.

It is said that one-piece has standard View If it contains only one numeric multiplier, standing in the first place, and each product of the same variables in it is represented by the degree. The numerical factor is unobed, noted in standard form, called single coefficient . Degree of one-Ukrainian call the sum of the degrees of all its variables.

Definition 3.4. Polynomial call the amount of single-wing. Scheduled from which the polynomial is compiled, calledmembers of the polynomial .

Similar terms - unrocked in the polynomial - called similar members of the polynomial .

Definition 3.5. Polynomial standard view they call a polynomial in which all the components are recorded in a standard form and similar members are given.The degree of polynomial of the standard type They call the greatest of the degrees of universal in it.

For example, the standard type of the fourth degree is polynomial.

Actions on single-wing and polynomials

The amount and difference of polynomials can be converted to a polynomial of a standard species. When adding two polynomials, all their members are recorded and similar members are given. When subtracting the signs of all members of the subdued polynomial change to the opposite.

For example:

Members of the polynomial can be broken into groups and enter into brackets. Since it is identical conversion, reverse disclosure of brackets, then the following is set. rule of imprisonment in brackets: if the "Plus" sign is set in front of the brackets, then all members concluded in brackets are recorded with their signs; If the "minus" sign is installed in front of the brackets, then all members entered into the brackets are recorded with opposite signs.

For example,

Multiplication rule by polynomial: to multiply a polynomial per polynomial, each member of one polynomial multiplied to each member of the other polynomial and the resulting works are folded.

For example,

Definition 3.6. Polynomial from one variable degree call the expression of the view

where
- Any numbers called coefficients of polynomial , and
,- a non-negative number.

If a
, then coefficient call senior polynomial coefficient
Singoral
- his senior member , coefficient – free member .

If instead of a variable in polynomial
enter a valid number then resulting in a valid number
called multicoral value
for
.

Definition 3.7. Number callroot polynomial
, if a
.

Consider the division of the polynomial on the polynomial where
and - integers. Division is possible if the degree of polynomial-division
no less than the degree of polynomial-divider
, i.e
.

Divide the polynomial
on polynomial
,
- it means to find two such polynomials
and
to

At the same time a polynomial
degree
call polynomial-private ,
– residue ,
.

Note 3.2. If divisel
– NOT zero-polynomial, then division
on the
,
Always fulfilled, and the private and the residue is definitely determined.

Note 3.3. In the case when
at all , i.e

it is said that the polynomial
fucally divided(or sharing) on polynomial
.

The division of polynomials is performed similarly to the division of multivalued numbers: First, the senior member of the polynomial-division is divided into the elder member of the polynomial-divider, then the private from the division of these members, which will be a senior member of the polynomial-private, multiplious to the polynomial-divider and the resulting product is deducted from the polynomial-division . As a result, a polynomial is obtained - the first residue, which is divided into a polynomial-divider in the same way and find the second member of the polynomial-private. This process is continued until the zero residue is obtained or the degree of polynomial residue will be less than the degree of polynomial-divider.

When dividing the polynomial on bounce, you can use the Gorner Scheme.

Gorner scheme

Let it take to divide the polynomial

on bounce
. Denote by private from division as a polynomial

and the residue is . Value , polynomial coefficients
,
and residue we write in the following form:

In this scheme, each of the coefficients
,
,
, …,it turns out from the previous number of the bottom line by multiplying and adding to the resulting result of the appropriate number of the top row standing above the desired coefficient. If any degree there is no in the polynomial, the corresponding coefficient is zero. Identify the coefficients according to the shown scheme, write the private

and fission result if
,

or ,

if a
,

Theorem 3.1. In order for an inconspicuous fraction (

,

) was the root of the polynomial
with whole coefficients, it is necessary that the number was a free member divider , and number - Divider of the older coefficient .

Theorem 3.2. (Theorem Bezu ) Residue from division of polynomial
on bounce
equal to the value of the polynomial
for
, i.e
.

When dividing the polynomial
on bounce
we have equality

It is fair, in particular,
, i.e
.

Example 3.2.Divide by
.

Decision.Apply the Gunner scheme:

Hence,

Example 3.3.Divide by
.

Decision.Apply the Gunner scheme:

Hence,

Example 3.4.Divide by
.

Decision.

As a result, we get

Example 3.5.Split
on the
.

Decision.We will conduct the division of polynomials by the Stage:

Then get

Sometimes it is useful to represent a polynomial in the form of a two or several polynomials equal to it. This identical conversion is called decomposition of polynomial to multipliers . Consider the main ways of such decomposition.

Making a common factor for brackets. In order to decompose the polynomials on the factors by the method of making a common factor behind the brackets, it is necessary:

1) Find a general factor. To do this, if all the coefficients of the polynomial are integers, as the general factory coefficient consider the largest total divisor of all polynomial coefficients, and each variable that is part of all members of the polynomial, take the highest indicator, which it has in this polynomial;

2) find a private from dividing this polynomial to a common factor;

3) Write the product of the general factor and the received private.

Grouping members. With the decomposition of the polynomial on the multipliers of the grouping method, its members are divided into two or more groups with such a calculation so that each of them can be converted into a work, and the obtained works would have a common factor. After that, a method for making a general factor of newly transformed members is applied.

Applying formulas of abbreviated multiplication. In cases where the polynomial to be decomposed the factors, it has the formula of any formula of abbreviated multiplication, its decomposition on multipliers is achieved by using the appropriate formula written in a different order.

Let be

, then are the following formulas of abbreviated multiplication:





For :
If a odd ):
Binin Newton: where - the number of combinations from by .

Introduction of new subsidiary members. This method is that the polynomial is replaced by another polynomial, identically equal to it, but containing another number of members, by introducing two opposite members or replacing any member identically equal to the sum of such universions. The replacement is made with such a calculation so that the method of grouping members can be applied to the obtained polynomial.

Example 3.6..

Decision.All members of the polynomial contain a general multiplier
. Hence,.

Answer: .

Example 3.7.

Decision.We group separately members containing the coefficient , and members containing . By making general multipliers of groups for parentheses, we get: