Learning to bring polynomial to the standard form

Among the various expressions, which are considered in algebra, the amount of homorals occupy an important place. We give 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 amount of homorals is called polynomial. The components in the polynomial are called members of the polynomial. We are also unintently refer to the polynomials, counting is unintently by a polynomial consisting of one member.

For example, polynomial
\\ (8B ^ 5 - 2B \\ Cdot 7b ^ 4 + 3b ^ 2 - 8b + 0.25b \\ Cdot (-12) B + 16 \\)
You can simplify.

Imagine all the components in the form of standard species:
\\ (8B ^ 5 - 2B \\ CDOT 7B ^ 4 + 3B ^ 2 - 8B + 0.25B \\ CDOT (-12) B + 16 \u003d \\)
\\ (\u003d 8b ^ 5 - 14b ^ 5 + 3b ^ 2 -8b -3b ^ 2 + 16 \\)

We give such members in the resulting polynomial:
\\ (8b ^ 5 -14b ^ 5 + 3b ^ 2 -8b -3b ^ 2 + 16 \u003d -6b ^ 5 -8b + 16 \\)
It turned out a polynomial, all members of which are one-sided species, and there are no similar among them. Such polynomials are called polynomials of standard species.

Per the degree of polynomial The standard species take the largest of the degrees of its members. Thus, bicked \\ (12a ^ 2b - 7b \\) has a third degree, and three stages \\ (2b ^ 2 -7b + 6 \\) - the second.

Typically, members of the polynomials of a standard form containing one variable are placed in the order of decrease in its degree. For example:
\\ (5x - 18x ^ 3 + 1 + x ^ 5 \u003d x ^ 5 - 18x ^ 3 + 5x + 1 \\)

The sum of several polynomials can be converted (simplify) into a polynomial of a standard species.

Sometimes members of the polynomial need to be divided into groups by entering into each group in brackets. Since conclusion in brackets is a transformation, reverse disclosure of brackets, it is easy to formulate rules for disclosing brackets:

If the "+" sign is set in front of the brackets, the members enclosed in brackets are recorded with the same signs.

If the "-" sign is installed in front of the brackets, the members concluded in the brackets are recorded with opposite signs.

Transformation (simplification) of works of single-wing and polynomial

Using the distribution properties of multiplication, you can convert (simplify) into a polynomial, the product is unoblared and polynomial. For example:
\\ (9a ^ 2b (7a ^ 2 - 5ab - 4b ^ 2) \u003d \\)
\\ (\u003d 9a ^ 2b \\ Cdot 7a ^ 2 + 9a ^ 2B \\ CDOT (-5AB) + 9A ^ 2B \\ CDOT (-4B ^ 2) \u003d \\)
\\ (\u003d 63a ^ 4b - 45a ^ 3b ^ 2 - 36A \u200b\u200b^ 2b ^ 3 \\)

The work is unobed and the polynomial is identically equal to the amount of works of this single and each of the members of the polynomial.

This result is usually formulated as a rule.

To multiply unripe of a polynomial, you need to multiply this one is unknown for each of the members of the polynomial.

We have repeatedly used this rule for multiplication by the amount.

The product of polynomials. Transformation (simplification) works of two polynomials

In general, the product of two polynomials is identically equal to the amount of the work of each member of one polynomial and each member of the other.

Usually enjoy the following rule.

To multiply the polynomial to the polynomial, each member of one polynomial is multiplied by each member of the other and folded the obtained works.

Formulas of abbreviated multiplication. Squares of the amount, difference and difference of squares

With some expressions in algebraic transformations, it is necessary to deal more often than with others. Perhaps the most common expressions \\ ((a + b) ^ 2, \\; (a - b) ^ 2 \\) and \\ (a ^ 2 - b ^ 2 \\), i.e., the sum of the sum, the square of the difference and Square differences. You noticed that the names of the specified expressions are not over, so, for example, \\ ((a + b) ^ 2 \\) is, of course, not just the square of the amount, and the square of the sum A and B. However, the square of the amount A and B is not so often, as a rule, instead of letters a and b, it turns out to be different, sometimes quite complex expressions.

Expressions \\ ((a + b) ^ 2, \\; (a - b) ^ 2 \\) It is not difficult to convert (simplify) into polynomials of a standard species, in fact, you have already met with such a task when multiplying polynomials:
\\ ((a + b) ^ 2 \u003d (a + b) (a + b) \u003d a ^ 2 + AB + Ba + B ^ 2 \u003d \\)
\\ (\u003d a ^ 2 + 2ab + b ^ 2 \\)

The obtained identities are useful to remember and apply without intermediate calculations. A brief verbal wording helps this.

\\ ((a + b) ^ 2 \u003d a ^ 2 + b ^ 2 + 2ab \\) - the sum of the sum is equal to the sum of the squares and the doubled work.

\\ ((a - b) ^ 2 \u003d a ^ 2 + b ^ 2 - 2ab \\) - the square of the difference is equal to the sum of the squares without a double product.

\\ (a ^ 2 - b ^ 2 \u003d (a - b) (a + b) \\) - the difference of squares is equal to the product of the difference in the amount.

These three identities allow in transformations to replace their left parts with the right and back - right parts left. The most difficult at the same time - see the appropriate expressions and understand how variables A and B are replaced. Consider several examples of using the formulas of abbreviated multiplication.

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 of general educational institutions / A. Mordkovich. - 17th ed., Extras - M.: Mnemozina, 2013. - 175 p.: Il. ISBN 978-5-346-02432-3.
Algebra and began 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.

- polynomials. In this article we will present all the initial and necessary information about the polynomials. To them, firstly, the definition of a polynomial with the accompanying definitions of members of the polynomial, in particular, a free member and similar members. Secondly, we will dwell on the polynomials of the standard species, we will give the appropriate definition and give their examples. Finally, we introduce the determination of the degree of polynomial, we will understand how to find it, and let's say about the coefficients of members of the polynomial.

Navigating page.

The polynomials and its members - definitions and examples

In grade 7, the polynomials are studied immediately after one-wing, it is understandable, since definition of polynomial Gives via unrocked. Let us give this definition explaining what a polynomial is.

Definition.

Polynomial - this is the sum of one-pans; Single is considered a private case of a polynomial.

The recorded definition allows you to cite how many examples of polynomials. Any of single-wing 5, 0, -1, x, 5 · a · b 3, x 2 · 0,6 · x · (-2) · y 12, and the like. is a polynomial. Also by definition 1 + x, a 2 + b 2 and are polynomials.

For the convenience of describing polynomials, the definition of a member of the polynomial is introduced.

Definition.

Members of the polynomial - This is the components of the polynomial shake.

For example, a polynomial 3 · x 4 -2 · x · y + 3 - y 3 consists of four members: 3 · x 4, -2 · x · y, 3 and -y 3. Single is considered a polynomial consisting of one member.

Definition.

Polynomials that consist of two and three members have special names - binomial and trinomial respectively.

So x + y is twisted, and 2 · x 3 · q-q · x · x + 7 · b - threest.

At school most often have to work with linear bounce a · x + b, where a and b - some numbers, and x - variable, as well as square threestyle A · x 2 + b · X + C, where A, B and C are some numbers, and x is a variable. Here are examples of linear bouncements: x + 1, x · 7.2-4, but examples of square three-stages: x 2 + 3 · x-5 and .

The polynomials in their record may have similar components. For example, in a polynomial 1 + 5 · x-3 + y + 2 · X, similar terms are 1 and -3, as well as 5 · x and 2 · x. They have their own special name - similar members of the polynomial.

Definition.

Similar members of the polynomial Call similar components in the polynomial.

In the previous example 1 and -3, as well as steam 5 · x and 2 · x are similar members of the polynomial. In polynomials having such members, it is possible to simplify their species to bring such members.

Polynomial standard view

For polynomials, as well as for single-wing, there is a so-called standard view. Let's voice the corresponding definition.

Based this definition, You can cite examples of standard type polynomials. So polynomials 3 · x 2 -x · y + 1 and recorded in standard form. And expressions 5 + 3 · x 2 -x 2 + 2 · x · z and x + x · x · z and x · x · y 3 · x · z 2 + 3 · z are not polynomials of the standard species, since the first of these contains similar members 3 · x 2 and -x 2, and in the second - single-wing x · y 3 · x · z 2, the type of which is different from the standard one.

Note that, if necessary, you can always bring a polynomial to the standard form.

The polynomials of the standard species include another concept - the concept of a free member of the polynomial.

Definition.

Free member of the polynomial They call a member of the standard type of numerous species without the letter.

In other words, if there is a number in the recording of a standard view, then it is called a free member. For example, 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?

Another important concomitant definition is to determine the degree of polynomial. First, we define the degree of polynomial of the standard species, this definition is based on the degrees of single-wing in its composition.

Definition.

The degree of polynomial of the standard type - This is the largest of the degrees of the recording of homorals.

We give examples. The degree of polynomial 5 · x 3 -4 is 3, since the composition of 5 · x 3 and -4 included in its composition have a degree 3 and 0, respectively, there are 3 of these numbers, it is the degree of polynomial by definition. And the degree of polynomial 4 · x 2 · y 3 -5 · x 4 · y + 6 · x It is equal to the largest of the numbers 2 + 3 \u003d 5, 4 + 1 \u003d 5 and 1, that is, 5.

Now find out how to find a degree of polynomial of an arbitrary type.

Definition.

Degree of polynomial arbitrary type They call the degree of the corresponding polynomial of the standard species.

So, if the polynomial is not recorded not in the standard form, and it is required to find its degree, then you need to bring the original polynomial to the standard form, and find the degree of polynomial obtained - it will be the desired. Consider the solution of the example.

Example.

Find a degree of polynomial 3 · a 12 -2 · a · b · c · a · c · b + y 2 · z 2 -2 · a 12 -a 12.

Decision.

First you need to submit a polynomial in the standard form:
3 · a 12 -2 · a · b · c · a · c · b + y 2 · z 2 -2 · a 12 -a 12 \u003d \u003d (3 · A 12 -2 · A 12 -A 12) - 2 · (a · a) · (b · b) · (c · c) + y 2 · z 2 \u003d \u003d -2 · a 2 · b 2 · C 2 + Y 2 · Z 2.

The resulting polynomial of the standard species includes two single-pillars -2 · a 2 · b 2 · C 2 and Y 2 · Z 2. We find them degrees: 2 + 2 + 2 \u003d 6 and 2 + 2 \u003d 4. Obviously, the largest of these degrees is equal to 6, it is by definition is the degree of polynomial -2 · a 2 · b 2 · C 2 + Y 2 · Z 2So, the degree of source polynomial., 3 · x and 7 polynomial 2 · x-0,5 · x · y + 3 · x + 7.

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 of general educational institutions / A. Mordkovich. - 17th ed., Extras - M.: Mnemozina, 2013. - 175 p.: Il. ISBN 978-5-346-02432-3.
Algebra and began 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.

In studying the topic of polynomials, it is worth mentioning that the polynomials are found both standard and not a standard species. In this case, the polynomial of the non-standard species can be caused by the standard form. Actually, this question will be disassembled in this article. Secure explanation with examples with a detailed step by step description.

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The meaning of bringing the polynomial to the standard form

A little deeper in the very concept, action - "bringing a polynomial to the standard form."

Polynomials, like any other expressions, it is possible to foreseally convert. As a result, we obtain the expression that are identically equal to the initial expression.

Definition 1.

Bring polynomial to standard - means the replacement of the original polynomial to the standard polynomial equal to it, obtained from the initial polynomial with the help of identical transformations.

Method of bringing polynomial to standard

We stroll on exactly which identical transformations Let's lead a polynomial to the standard form.

Definition 2.

According to the definition, each polynomial of a standard species consists of single-sided patterns and does not have similar members in its composition. The polynomial of the non-standard species may include unknown non-standard species and similar members. From the above, the rule tells the rule telling how to bring polynomial to the standard form:

first of all, the standard appearance is given to the standard components of the specified polynomial;
then brought similar members.

Examples and solutions

We will analyze in detail examples in which we give a polynomial to the standard form. We will follow the rule taken above.

Note that sometimes members of the polynomial in the initial state already have a standard appearance, and it remains only to bring similar members. It happens that after the first step of action, there are no such members, then we skip the second step. IN common cases It is necessary to make both actions from the rule above.

Example 1.

Set polynomials:

5 · x 2 · y + 2 · y 3 - x · y + 1 ,

0, 8 + 2 · a 3 · 0, 6 - b · a · b 4 · b 5,

2 3 7 · x 2 + 1 2 · y · x · (- 2) - 1 6 7 · x · x + 9 - 4 7 · x 2 - 8.

It is necessary to bring them to the standard form.

Decision

consider first a polynomial 5 · x 2 · y + 2 · y 3 - x · y + 1 : its members have a standard appearance, there are no such members, which means the polynomial is set in standard form, and no additional actions are required.

Now we will analyze the polynomial 0, 8 + 2 · a 3 · 0, 6 - b · a · b 4 · b 5. It consists of non-standard universal: 2 · a 3 · 0, 6 and - b · a · b 4 · b 5, i.e. We have the need to bring a polynomial to the standard form, for which the first action is converting universal to the standard form:

2 · a 3 · 0, 6 \u003d 1, 2 · a 3;

- b · a · b 4 · b 5 \u003d - a · b 1 + 4 + 5 \u003d - a · b 10, thus we get the following polynomial:

0, 8 + 2 · a 3 · 0, 6 - b · a · b 4 · b 5 \u003d 0, 8 + 1, 2 · a 3 - a · b 10.

In the resulting polynomial, all members are standard, such members do not have, it means our actions to bring the polynomial to the standard form are completed.

Consider the third set polynomial: 2 3 7 · x 2 + 1 2 · y · x · (- 2) - 1 6 7 · x · x + 9 - 4 7 · x 2 - 8

We give its members to the standard form and get:

2 3 7 · x 2 - x · y - 1 6 7 · x 2 + 9 - 4 7 · x 2 - 8.

We see that there are similar members as part of the polynomial, we will bring similar members:

2 3 7 · x 2 - x · y - 1 6 7 · x 2 + 9 - 4 7 · x 2 - 8 \u003d 2 3 7 · x 2 - 1 6 7 · x 2 - 4 7 · x 2 - x · Y + (9 - 8) \u003d x 2 · 2 3 7 - 1 6 7 - 4 7 - x · y + 1 \u003d x 2 · 17 7 - 13 7 - 4 7 - x · y + 1 \u003d \u003d x 2 · 0 - x · y + 1 \u003d x · y + 1

Thus, the specified polynomial 2 3 7 · x 2 + 1 2 · y · x · (- 2) - 1 6 7 · x · x + 9 - 4 7 · x 2 - 8 accepted the standard species - x · y + 1 .

Answer:

5 · x 2 · y + 2 · y 3 - x · y + 1 - the polynomial is set standard;

0, 8 + 2 · a 3 · 0, 6 - b · a · b 4 · b 5 \u003d 0, 8 + 1, 2 · a 3 - a · b 10;

2 3 7 · x 2 + 1 2 · y · x · (- 2) - 1 6 7 · x · x + 9 - 4 7 · x 2 - 8 \u003d - x · y + 1.

In many challenges, the action of bringing the polynomial to the standard appearance is intermediate when searching for an answer to asked question. Consider such an example.

Example 2.

The polynomial 11 - 2 3 Z 2 · Z + 1 3 · Z 5 · 3 - 0 is set. 5 · Z 2 + Z 3. It is necessary to bring it to with a standard form, to specify its degree and arrange the members of the specified polynomial in decreasing degrees of the variable.

Decision

We present the members of the specified polynomial to the standard form:

11 - 2 3 Z 3 + Z 5 - 0. 5 · Z 2 + Z 3.

The next step will give similar members:

11 - 2 3 Z 3 + Z 5 - 0. 5 · z 2 + z 3 \u003d 11 + - 2 3 · z 3 + z 3 + z 5 - 0, 5 · z 2 \u003d 11 + 1 3 · z 3 + z 5 - 0, 5 · z 2

We received a polynomial of the standard species, which gives us the ability to designate the degree of polynomial (equal to the greatest degree of its components of its homorals). Obviously, the desired degree is equal to 5.

It remains only to arrange members of descending degrees of variables. To this end, we will simply reset the members in the obtained polynomial of the standard type, taking into account the requirements. Thus, we get:

z 5 + 1 3 · z 3 - 0, 5 · z 2 + 11.

Answer:

11 - 2 3 · z 2 · z + 1 3 · z 5 · 3 - 0, 5 · z 2 + z 3 \u003d 11 + 1 3 · z 3 + z 5 - 0, 5 · z 2, with a degree of polynomial - five ; As a result of the location of the members of the polynomial in decreasing degrees of variables, the polynomial takes the form: z 5 + 1 3 · z 3 - 0, 5 · z 2 + 11.

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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 of different kind 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 to calculate the specific value of the polynomial at the specified numerical values 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.