Polynomial, its standard form, degree and coefficients of terms. Reducing polynomials to standard form. Typical tasks

We said that there are both standard and non-standard polynomials. In the same place, we noted that any polynomial lead to standard view ... In this article, we will first find out what the meaning of this phrase is. Next, we list the steps that allow you to transform any polynomial into a standard form. Finally, consider solutions to typical examples. We will describe the solutions in great detail in order to deal with all the nuances that arise when bringing polynomials to a standard form.

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What does it mean to bring a polynomial to a standard form?

First, you need to clearly understand what is meant by reducing a polynomial to a standard form. Let's figure it out.

Polynomials, like any other expression, can be subjected to identical transformations. As a result of performing such transformations, expressions are obtained that are identically equal to the original expression. So the implementation of certain transformations with polynomials of a non-standard form allows one to go to polynomials identically equal to them, but written already in the standard form. This transition is called the reduction of the polynomial to the standard form.

So, bring polynomial to standard form- this means replacing the original polynomial with an identically equal polynomial of the standard form obtained from the original by carrying out identical transformations.

How to bring a polynomial to a standard form?

Let's think about what transformations will help us to bring the polynomial to its standard form. We will start from the definition of a standard polynomial.

By definition, each member of a standard-form polynomial is a standard-form monomial, and a standard-form polynomial does not contain such members. In turn, polynomials written in a form other than the standard one can consist of monomials in a non-standard form and can contain similar terms. Hence the following rule logically follows, explaining how to bring a polynomial to standard form:

• first you need to reduce to the standard form the monomials that make up the original polynomial,
• and then perform the casting of similar members.

As a result, a polynomial of the standard form will be obtained, since all its members will be written in the standard form, and it will not contain such members.

Examples, solutions

Let's consider examples of reducing polynomials to the standard form. When deciding, we will follow the steps dictated by the rule from the previous paragraph.

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

Example.

Represent the polynomials in the 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 .

Solution.

All terms of the polynomial 5 · x 2 · y + 2 · y 3 −x · y + 1 are written in the standard form, it has no similar terms, therefore, this polynomial is already represented in the standard form.

Go to the next polynomial 0.8 + 2 a 3 0.6 − b a b 4 b 5... Its form is not standard, as evidenced by the terms 2 · a 3 · 0.6 and −b · a · b 4 · b 5 are not standard form. Let's represent it in the standard form.

At the first stage of bringing the original polynomial to a standard form, we need to present all its members in a standard form. Therefore, we reduce the monomial 2 a 3 0.6 to the standard form, we have 2 a 3 0.6 = 1.2 a 3, after which the monomial −b a b 4 b 5, we have −b a b 4 b 5 = −a b 1 + 4 + 5 = −a b 10... Thus, . In the resulting polynomial, all terms are written in a standard form, moreover, it is obvious that there are no similar terms in it. Consequently, this completes the reduction of the original polynomial to the standard form.

It remains to present the last of the given polynomials in a standard form. After bringing all its members to the standard form, it will be written as ... It has similar members, so you need to cast such members:

So the original polynomial took the standard form −x · y + 1.

Answer:

5 x 2 y + 2 y 3 −x y + 1 - already in the standard form, 0.8 + 2 a 3 0.6 − b a b 4 b 5 = 0.8 + 1.2 a 3 −a b 10, .

Often, bringing a polynomial to a standard form is only an intermediate stage in answering the question posed in the problem. For example, finding the degree of a polynomial assumes its preliminary representation in a standard form.

Example.

Give the polynomial to the standard form, indicate its degree and arrange the terms in decreasing powers of the variable.

Solution.

First, we bring all the terms of the polynomial to the standard form: .

Now we give similar members:

So we brought the original polynomial to the standard form, this allows us to determine the degree of the polynomial, which is equal to the largest degree of the monomials included in it. Obviously, it is equal to 5.

It remains to arrange the terms of the polynomial in decreasing powers of the variables. To do this, you just need to rearrange the terms in the resulting polynomial of the standard form, taking into account the requirement. The term z 5 has the highest degree, the degrees of the terms −0.5 · z 2 and 11 are 3, 2, and 0, respectively. Therefore, a polynomial with terms in decreasing degrees of the variable will have the form .

Answer:

The degree of the polynomial is 5, and after the arrangement of its terms in decreasing degrees of the variable, it takes the form .

Bibliography.

• Algebra: study. for 7 cl. general education. institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 17th ed. - M.: Education, 2008 .-- 240 p. : ill. - ISBN 978-5-09-019315-3.
• A. G. Mordkovich Algebra. 7th grade. At 2 pm Part 1. Textbook for students of educational institutions / A. G. Mordkovich. - 17th ed., Add. - M .: Mnemozina, 2013 .-- 175 p .: ill. ISBN 978-5-346-02432-3.
• Algebra and the beginning of mathematical analysis. Grade 10: textbook. for general education. institutions: basic and profile. levels / [Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; ed. A. B. Zhizhchenko. - 3rd ed. - M .: Education, 2010.- 368 p. : ill. - ISBN 978-5-09-022771-1.
• Gusev V.A., Mordkovich A.G. Mathematics (manual for applicants to technical schools): Textbook. manual. - M .; Higher. shk., 1984.-351 p., ill.

Among the various expressions that are considered in algebra, the 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 the polynomial are called the terms of the polynomial. Monomials are also referred to as polynomials, considering a monomial to be a polynomial consisting of one term.

For example, the polynomial
\ (8b ^ 5 - 2b \ cdot 7b ^ 4 + 3b ^ 2 - 8b + 0.25b \ cdot (-12) b + 16 \)
can be simplified.

We 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 of whose members are monomials of the standard form, and there are no similar ones among them. Such polynomials are called polynomials of the standard form.

Per polynomial degree of the standard form take the largest of the degrees of its members. So, the binomial \ (12a ^ 2b - 7b \) has the third degree, and the trinomial \ (2b ^ 2 -7b + 6 \) - the second.

Usually, the members of polynomials of the standard form containing one variable are arranged in descending order of the exponents of its exponent. For example:
\ (5x - 18x ^ 3 + 1 + x ^ 5 = x ^ 5 - 18x ^ 3 + 5x + 1 \)

The sum of several polynomials can be converted (simplified) into a polynomial of the standard form.

Sometimes the members of a polynomial need to be divided into groups, enclosing each group in parentheses. Since parenthesis is the reverse of parenthesis expansion, it is easy to formulate parenthesis expansion rules:

If the "+" sign is placed in front of the brackets, then the members enclosed in brackets are written with the same signs.

If the “-” sign is placed in front of the brackets, then the members enclosed in brackets are written with opposite signs.

Transformation (simplification) of the product of a monomial and a polynomial

Using the distribution 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 members of the polynomial.

This result is usually formulated as a rule.

To multiply a monomial by a polynomial, you need to multiply this monomial by each of the members of the polynomial.

We have already used this rule for multiplying by a sum many times.

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 member of one polynomial and each member 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

Some expressions in algebraic transformations have to be dealt with more often than others. Perhaps the most common expressions \ ((a + b) ^ 2, \; (a - b) ^ 2 \) and \ (a ^ 2 - b ^ 2 \), that is, the square of the sum, the square of the difference, and difference of squares. You have noticed that the names of these expressions seem to be incomplete, so, 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 is not so common, as a rule, instead of the letters a and b, it contains different, sometimes rather complex expressions.

Expressions \ ((a + b) ^ 2, \; (a - b) ^ 2 \) are easy to transform (simplify) into polynomials of the standard form, in fact, you have already encountered this task when multiplying polynomials:
\ ((a + b) ^ 2 = (a + b) (a + b) = a ^ 2 + ab + ba + b ^ 2 = \)
\ (= a ^ 2 + 2ab + b ^ 2 \)

The obtained identities are useful to remember and apply 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 doubled product.

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

\ (a ^ 2 - b ^ 2 = (a - b) (a + b) \) - the difference of the squares is equal to the product of the difference by the sum.

These three identities allow in transformations to replace their left-hand sides with the right ones and vice versa - the right-hand sides with the left ones. The most difficult thing is to see the corresponding expressions and understand what replaces the variables a and b in them. Let's look at some examples of using abbreviated multiplication formulas.

In studying the topic of polynomials, it is worth mentioning separately that polynomials are found in both standard and non-standard forms. In this case, a non-standard polynomial can be reduced to a standard form. Actually, this question will be analyzed in this article. Let's fix the explanations with examples with a detailed step-by-step description.

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The meaning of reducing a polynomial to a standard form

Let's go a little deeper into the concept itself, the action is "bringing a polynomial to a standard form."

Polynomials, like any other expression, can be identically transformed. As a result, we get in this case expressions that are identically equal to the original expression.

Definition 1

Reduce polynomial to standard form- means replacing the original polynomial with an equal polynomial of the standard form, obtained from the original polynomial using identical transformations.

Method of reducing a polynomial to a standard form

We speculate on the topic of which identical transformations will bring the polynomial to the standard form.

Definition 2

According to the definition, each polynomial of the standard form consists of monomials of the standard form and does not have such members in its composition. A polynomial of a non-standard form can include non-standard monomials and similar members. From what has been said, a rule is naturally derived that says how to bring a polynomial to a standard form:

• first of all, the monomials that make up a given polynomial are reduced to the standard form;
• then similar members are cast.

Examples and solutions

Let us analyze in detail examples in which we bring the polynomial to its standard form. We will follow the rule deduced above.

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

Example 1

Polynomials are given:

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.

Solution

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

Now let's analyze the polynomial 0, 8 + 2 · a 3 · 0, 6 - b · a · b 4 · b 5. It includes non-standard monomials: 2 a 3 0, 6 and - b a b 4 b 5, i.e. we need to bring the polynomial to the standard form, for which the first action we transform the monomials into the standard form:

2 a 3 0, 6 = 1, 2 a 3;

- b a b 4 b 5 = - a b 1 + 4 + 5 = - a b 10, so we get the following polynomial:

0.8 + 2 a 3 0, 6 - b a b 4 b 5 = 0.8 + 1, 2 a 3 - a b 10.

In the resulting polynomial, all the members are standard, there are no such members, which means that our steps to bring the polynomial to the standard form are completed.

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

Let's bring 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 in 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 = = 2 3 7 x 2 - 1 6 7 x 2 - 4 7 x 2 - x Y + (9 - 8) = = x 2 2 3 7 - 1 6 7 - 4 7 - x y + 1 = = x 2 17 7 - 13 7 - 4 7 - x y + 1 = = x 2 0 - x y + 1 = x y + 1

Thus, the given polynomial 2 3 7 x 2 + 1 2 y x (- 2) - 1 6 7 x x + 9 - 4 7 x 2 - 8 takes the standard form - x y + 1 ...

Answer:

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

0.8 + 2 a 3 0.6 - b a b 4 b 5 = 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 = - x y + 1.

In many problems, the action of reducing a polynomial to a standard form is an intermediate one when looking for an answer to the question asked... Consider this example.

Example 2

A polynomial 11 - 2 3 z 2 z + 1 3 z 5 3 - 0 is given. 5 z 2 + z 3. It is necessary to bring it to a standard form, indicate its degree and arrange the terms of the given polynomial in decreasing powers of the variable.

Solution

Let us bring the terms of the given polynomial to the standard form:

11 - 2 3 z 3 + z 5 - 0. 5 z 2 + z 3.

The next step is to bring similar members:

11 - 2 3 z 3 + z 5 - 0. 5 z 2 + z 3 = 11 + - 2 3 z 3 + z 3 + z 5 - 0.5 z 2 = = 11 + 1 3 z 3 + z 5 - 0.5 z 2

We have obtained a polynomial of the standard form, which allows us to denote the degree of the polynomial (equal to the greatest degree of its constituent monomials). Obviously, the required degree is 5.

It remains only to arrange the terms in decreasing powers of the variables. For this purpose, we simply rearrange the positions of the terms in the resulting polynomial of the standard form, taking into account the requirement. 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 = 11 + 1 3 z 3 + z 5 - 0.5 z 2, while the degree of the polynomial - 5 ; as a result of the arrangement of the terms of the polynomial in decreasing powers of the variables, the polynomial takes the form: z 5 + 1 3 · z 3 - 0.5 · z 2 + 11.

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For example, expressions:

a - b + c, x 2 - y 2 , 5x - 3y - z- polynomials

The monomials that make up a polynomial are called members of the polynomial... Consider a polynomial:

7a + 2b - 3c - 11

expressions: 7 a, 2b, -3c and -11 are members of the polynomial. Pay attention to the -11 member, it does not contain a variable, such members consisting only of a number are called free.

It is generally accepted that any monomial is special case polynomial, consisting of one term. In this case, a monomial is the name for a polynomial with one term. For polynomials consisting of two and three members, there are also special names - two-term and three-term, respectively:

7a- monomial

7a + 2b- binomial

7a + 2b - 3c- three-member

Similar members

Similar members- monomials included in a polynomial, which differ from each other only by a coefficient, sign, or do not differ at all (opposite monomials can also be called similar). For example, in a polynomial:

3a 2 b

+

5abc 2

+

2a 2 b

-

7abc 2

-

2a 2 b

members 3 a 2 b, 2a 2 b and 2 a 2 b, as well as terms 5 abc 2 and -7 abc 2 are similar members.

Bringing similar members

If a polynomial contains similar terms, then it can be reduced to more simple mind by combining similar members into one. This action is called bringing similar members... First of all, we put all such members in brackets separately:

(3a 2 b + 2a 2 b - 2a 2 b) + (5abc 2 - 7abc 2)

To combine several similar monomials into one, you need to add their coefficients, and leave the letter factors unchanged:

((3 + 2 - 2)a 2 b) + ((5 - 7)abc 2) = (3a 2 b) + (-2abc 2) = 3a 2 b - 2abc 2

The reduction of similar terms is the operation of replacing the algebraic sum of several similar monomials with one monomial.

Standard polynomial

Standard polynomial is a polynomial, all members of which are monomials of the standard form, among which there are no similar members.

To bring a polynomial to a standard form, it is enough to make a reduction of similar terms. For example, imagine the expression as a standard polynomial:

3xy + x 3 - 2xy - y + 2x 3

First, let's find similar members:

If all members of a polynomial of the standard form contain the same variable, then its members are usually arranged from higher degree to lower degree. The free term of the polynomial, if any, is put on last place- on right.

For example, the polynomial

3x + x 3 - 2x 2 - 7

should be written like this:

x 3 - 2x 2 + 3x - 7

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

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Guys, you have already studied monomials in the topic: Standard monomial form. Definitions. Examples. Let's review the basic definitions.

Monomial- an expression consisting of the product of numbers and variables. Variables can be raised to natural degrees. The monomial does not contain any other actions than multiplication.

Standard type of monomial- such a form when the coefficient (numerical factor) comes first, followed by the degrees of various variables.

Similar monomials Are either identical monomials or monomials that differ from each other by a coefficient.

Polynomial concept

A polynomial, like a monomial, is a generalized name mathematical expressions a certain kind. We have encountered such generalizations before. For example, "sum", "product", "exponentiation". When we hear the "difference of numbers", the thought of multiplication or division will never occur to us. Likewise, a polynomial is an expression of a strictly defined kind.

Definition of a polynomial

Polynomial is the sum of monomials.

The monomials that make up a polynomial are called members of the polynomial... If there are two terms, then we are dealing with a binomial, if three, then with a trinomial. If they say more terms, it is a polynomial.

Examples of polynomials.

1) 2ab + 4cd (binomial);

2) 4ab + 3cd + 4x (trinomial);

3) 4a 2 b 4 + 4c 8 d 9 + 2xy 3;

3c 7 d 8 - 2b 6 c 2 d + 7xy - 5xy 2.

Let's take a close look at the last expression. By definition, a polynomial is the sum of monomials, but in the last example, we not only add, but also subtract monomials.
To be clear, let's look at a small example.

Let's write the expression a + b - c(we agree that a ≥ 0, b ≥ 0 and c ≥0) and answer the question: is it the sum or the difference? It is hard to say.
Indeed, if you rewrite the expression as a + b + (-c), we get the sum of two positive and one negative terms.
If you look at our example, then we are dealing precisely with the sum of monomials with coefficients: 3, - 2, 7, -5. In mathematics, there is the term "algebraic sum". Thus, the definition of a polynomial means "algebraic sum".

But the notation of the form 3a: b + 7c is not a polynomial because 3a: b is not a monomial.
The notation of the form 3b + 2a * (c 2 + d) is also not a polynomial, since 2a * (c 2 + d) is not a monomial. If you expand the brackets, then the resulting expression will be a polynomial.
3b + 2a * (c 2 + d) = 3b + 2ac 2 + 2ad.

The degree of the polynomial is the highest degree of its members.
The polynomial a 3 b 2 + a 4 has the fifth degree, since the degree of the monomial a 3 b 2 is 2 + 3 = 5, and the degree of the monomial a 4 is 4.

Standard form of a polynomial

A polynomial that does not have such terms and is written in decreasing order of degrees of the terms of the polynomial is a polynomial of the standard form.

The polynomial is brought to a standard form in order to remove unnecessary cumbersome writing and to simplify further actions with it.

Indeed, why, for example, write the long expression 2b 2 + 3b 2 + 4b 2 + 2a 2 + a 2 + 4 + 4, when it can be written shorter than 9b 2 + 3a 2 + 8.

To bring a polynomial to a standard form, you need:
1.to bring all its members to a standard form,
2. add up similar (identical or with a different numerical coefficient) terms. This procedure is often called bringing similar.

Example.
Bring the polynomial aba + 2y 2 x 4 x + y 2 x 3 x 2 + 4 + 10a 2 b + 10 to its standard form.

Solution.

a 2 b + 2 x 5 y 2 + x 5 y 2 + 10a 2 b + 14 = 11a 2 b + 3 x 5 y 2 + 14.

Let us determine the degrees of the monomials included in the expression, and arrange them in descending order.
11a 2 b has the third degree, 3 x 5 y 2 has the seventh degree, 14 - zero degree.
This means that in the first place we will put 3 x 5 y 2 (7 degree), in the second - 12a 2 b (3 degree) and in the third - 14 (zero degree).
As a result, we get a polynomial of the standard form 3x 5 y 2 + 11a 2 b + 14.

Examples for independent solution

Reduce polynomials to standard form.

1) 4b 3 aa - 5x 2 y + 6ac - 2b 3 a 2 - 56 + ac + x 2 y + 50 * (2 a 2 b 3 - 4x 2 y + 7ac - 6);

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

3) 4х 2 + 5bс - 6а - 24bс + хх 4 x (5х 6 - 19bс - 6а);

4) 7abc 2 + 5acbc + 7ab 2 - 6bab + 2cabc (14abc 2 + ab 2).