Bringing polynomials to standard form. Typical tasks. Polynomial and its standard species

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.

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The polynomials and its members - definitions and examples

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

Definition 1.

Polynomial It is considered the amount of homorals, and unrochet himself is private case 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 transformation and solutions, it is necessary to find and bring similar components. 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 driven standard. 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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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.

Lesson on the topic: "The concept and definition of a polynomial. Standard type of polynomial"

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Guys, you have already studied unknown in the subject: Standard species is unobed. Definitions. Examples. Let's repeat the basic definitions.

Monomial - expression consisting of the product of numbers and variables. Variables can be elevated to a natural extent. Unrochene does not contain any other actions except multiplication.

Standard view of Singochlenna - This kind of when the coefficient is in the first place (numerical factor), and the degree of various variables.

Similar homoral - These are either the same shake, or are unrocked, which differ from each other by the coefficient.

The concept of polynomial

The polynomial, as is unripe, is a generalized name. mathematical expressions a certain species. We have already encountered such generalizations earlier. For example, the "amount", "work", "exercise to the degree". When we hear the "difference of numbers", we will not have a thought of multiplication or division. Also, the polynomial is a strictly defined expression.

Definition of polynomial

Polynomial - This is the sum of one-wing.

Scheduled in the composition of the polynomial are called members of the polynomial. If the terms are two, then we are dealing with twisted, they have three, then with three stuck. If the components say more - a polynomial.

Examples of polynomials.

1) 2AB + 4SD (bounce);

2) 4AB + 3CD + 4X (three-shred);

3) 4A 2 B 4 + 4C 8 D 9 + 2XU 3;

3C 7 D 8 - 2B 6 C 2 D + 7XU - 5XY 2.

Let's see carefully for the latest expression. By definition, this is a polynomial - the amount of single-wing, but in the last example we not only fold, but we will deduct unrocked.
To make clarity Consider a small example.

We write expression a + b - with (agree that a ≥ 0, B ≥ 0 and C ≥0) And answer the question: is this amount or difference? It is hard to say.
Really if rewritten expression like a + b + (-s), we will receive the sum of two positive and one negative terms.
If you look at our example, we are dealing precisely with the sum of one-wing with the coefficients: 3, - 2, 7, -5. In mathematics there is a term "algebraic amount". Thus, in the definition of the polynomial, there is a "algebraic amount".

But the view of the type 3a: B + 7C is not a polynomial because 3a: b is not universal.
It is not a polynomial and record of the form 3b + 2a * (C 2 + D), since 2a * (C 2 + D) - not single. If you reveal the brackets, the resulting expression will be a polynomial.
3b + 2a * (C 2 + d) \u003d 3b + 2as 2 + 2Ad.

Degree of polynomial is the highest degree of its members.
The polynomial A 3 B 2 + A 4 has the fifthous degree, since the degree is universal and 3 B 2 is 2 + 3 \u003d 5, and the degree of universal a 4 is 4 equal to 4.

Standard type of polynomial

A polynomial that does not have such members and recorded in descending order of the degrees of the members of the polynomial is a polynomial of a standard species.

The polynomials lead to a standard form, to remove excessive bulkness of writing and simplify further actions with it.

Indeed, why for example, writing a long expression 2b 2 + 3B 2 + 4B 2 + 2a 2 + A 2 + 4 + 4, when it can be written in short 9b 2 + 3a 2 + 8.

To bring a polynomial to the standard form, it is necessary:
1. Create all its members to the standard form,
2. Compose similar (identical or different numerical coefficient) members. This procedure is often called by bringing similar.

Example.
Create a polynomial ABA + 2U 2 x 4 x + in 2 x 3 x 2 + 4 + 10a 2 b + 10 to standard form.

Decision.

a 2 b + 2 x 5 in 2 + x 5 in 2 + 10a 2 b + 14 \u003d 11a 2 b + 3 x 5 in 2 + 14.

We define the degrees of universions that are part of the expression, and put them in descending order.
11A 2 B has a third degree, 3 x 5 in 2 has a seventh degree, 14 - zero degree.
So, in the first place we put 3 x 5 in 2 (7 degree), on the second - 12a 2 B (3 degree) and on the third - 14 (zero degree).
As a result, we obtain the polynomial of the standard type 3x 5 in 2 + 11a 2 b + 14.

Examples for self-decisions

Lead to standard polynomials.

1) 4B 3 AA - 5x 2 y + 6as - 2B 3 A 2 - 56 + AC + X 2 y + 50 * (2 A 2 B 3 - 4x 2 y + 7as - 6);

2) 6a 5 b + 3x 2 y + 45 + x 2 y + AB - 40 * (6a 5 b + 4h + AB + 5);

3) 4ach 2 + 5BC - 6A - 24BC + HX 4 X (5H 6 - 19BC - 6A);

4) 7ABC 2 + 5Asbs + 7Ab 2 - 6BB + 2SABS (14ABC 2 + AB 2).

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;

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

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:

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: