Polynomial, its standard form, degree and coefficients of terms. Online calculator. Simplifying a polynomial. Multiplying polynomials

We said that there are both standard and non-standard polynomials. There we noted that anyone can bring the polynomial to standard form. In this article, we will first find out what meaning this phrase carries. Next we list the steps to convert any polynomial into standard form. Finally, let's look at solutions to typical examples. We will describe the solutions in great detail in order to understand all the nuances that arise when reducing polynomials to standard form.

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

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

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

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

How to reduce a polynomial to standard form?

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

By definition, every term of a polynomial of standard form is a monomial of standard form, and a polynomial of standard form contains no similar terms. 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. This logically follows the following rule, which explains how to reduce a polynomial to standard form:

• first you need to bring to standard form the monomials that make up the original polynomial,
• then perform the reduction of similar terms.

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

Examples, solutions

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

Let us note here that sometimes all the terms of a polynomial are immediately written in standard form; in this case, it is enough to just give similar terms. Sometimes, after reducing the terms of a polynomial to a standard form, there are no similar terms, therefore, the stage of bringing similar terms is omitted in this case. In general, you have to do both.

Example.

Present the 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 .

Solution.

All terms of the polynomial 5·x 2 ·y+2·y 3 −x·y+1 are written in standard form; it does not have similar terms, therefore, this polynomial is already presented in standard form.

Let's move on 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 of a non-standard form. Let's present it in standard form.

At the first stage of bringing the original polynomial to standard form, we need to present all its terms in standard form. Therefore, we reduce the monomial 2·a 3 ·0.6 to standard form, we have 2·a 3 ·0.6=1.2·a 3 , after which we take 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 standard form; moreover, it is obvious that there are no similar terms in it. Consequently, this completes the reduction of the original polynomial to standard form.

It remains to present the last of the given polynomials in standard form. After bringing all its members to standard form, it will be written as . It has similar members, so you need to cast similar 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 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 step in answering the posed question of the problem. For example, finding the degree of a polynomial requires its preliminary representation in standard form.

Example.

Give a polynomial to the standard form, indicate its degree and arrange the terms in descending degrees of the variable.

Solution.

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

Now we present similar terms:

So we brought the original polynomial to a standard form, this allows us to determine the degree of the polynomial, which is equal to the highest 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 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 equal to 3, 2 and 0, respectively. Therefore, a polynomial with terms arranged in decreasing powers of the variable will have the form .

Answer:

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

References.

• Algebra: textbook for 7th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
• Mordkovich A. G. Algebra. 7th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
• Algebra and the beginning of mathematical analysis. 10th grade: textbook. for general education institutions: basic and profile. levels / [Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; edited by A. B. Zhizhchenko. - 3rd ed. - M.: Education, 2010.- 368 p. : ill. - ISBN 978-5-09-022771-1.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

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

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The meaning of reducing a polynomial to standard form

Let's delve a little deeper into the concept itself, the action - “bringing a polynomial to a standard form.”

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

Definition 1

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

A method for reducing a polynomial to standard form

Let's speculate on the topic of exactly what identity transformations will lead the polynomial to the standard form.

Definition 2

According to the definition, each polynomial of standard form consists of monomials of standard form and does not contain similar terms. A polynomial of a non-standard form may include monomials of a non-standard form and similar terms. From the above, a rule is naturally deduced about how to reduce a polynomial to a standard form:

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

Examples and solutions

Let us examine in detail examples in which we reduce the polynomial to standard form. We will follow the rule derived above.

Note that sometimes the terms of a polynomial in the initial state already have a standard form, and all that remains is to bring similar terms. It happens that after the first step of actions there are no such terms, then we skip the second step. In general 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 a standard form.

Solution

Let's first consider the polynomial 5 x 2 y + 2 y 3 − x y + 1 : its members have a standard form, there are no similar terms, which means the polynomial is specified in a standard form, and no additional actions are required.

Now let's look at 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 standard form, for which the first step is to transform the monomials into 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 , thus we obtain 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 terms are standard, there are no similar terms, which means our actions to bring the polynomial to 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 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 the polynomial contains similar members, let’s 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 intermediate when searching for an answer to a given question. Let's consider this example.

Example 2

The 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 a given polynomial in descending degrees of the variable.

Solution

Let us reduce 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 present similar terms:

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 standard form, which allows us to designate the degree of the polynomial (equal to the highest degree of its constituent monomials). Obviously, the required degree is 5.

All that remains is to arrange the terms in decreasing powers of the variables. For this purpose, we simply rearrange the terms in the resulting polynomial of 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 arranging the terms of the polynomial in decreasing powers of the variables, the polynomial will take the form: z 5 + 1 3 · z 3 - 0, 5 · z 2 + 11.

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In this lesson, we will recall the basic definitions of this topic and consider some typical problems, namely, reducing a polynomial to a standard form and calculating a numerical value for given values ​​of variables. We will solve several examples in which reduction to a standard form will be used to solve various types of problems.

Subject:Polynomials. Arithmetic operations on monomials

Lesson:Reducing a polynomial to standard form. Typical tasks

Let us recall the basic definition: a polynomial is the sum of monomials. Each monomial that is part of a polynomial as a term is called its member. For example:

Binomial;

Polynomial;

Binomial;

Since a polynomial consists of monomials, the first action with a polynomial follows from here - you need to bring all monomials to a standard form. Let us remind you that to do this you need to multiply all the numerical factors - get a numerical coefficient, and multiply the corresponding powers - get the letter part. In addition, let us pay attention to the theorem about the product of powers: when powers are multiplied, their exponents add up.

Let's consider an important operation - reducing a polynomial to standard form. Example:

Comment: to bring a polynomial to a standard form, you need to bring all the monomials included in its composition to a standard form, after which, if there are similar monomials - and these are monomials with the same letter part - perform actions with them.

So, we looked at the first typical problem - bringing a polynomial to a standard form.

The next typical task is to calculate the specific value of a polynomial for given numerical values ​​of its variables. Let's continue to look at the previous example and set the values ​​of the variables:

Comment: recall that one to any natural power is equal to one, and zero to any natural power is equal to zero, in addition, recall that when multiplying any number by zero, we get zero.

Let's look at a number of examples of typical operations of reducing a polynomial to a standard form and calculating its value:

Example 1 - bring to standard form:

Comment: the first step is to bring the monomials to the standard form, you need to bring the first, second and sixth; second action - we bring similar terms, that is, we perform the given arithmetic operations on them: we add the first with the fifth, the second with the third, we rewrite the rest without changes, since they have no similar ones.

Example 2 - calculate the value of the polynomial from example 1 given the values ​​of the variables:

Comment: when calculating, you should remember that a unit to any natural power is one; if it is difficult to calculate powers of two, you can use the table of powers.

Example 3 - instead of an asterisk, put a monomial such that the result does not contain a variable:

Comment: regardless of the task, the first action is always the same - bring the polynomial to a standard form. In our example, this action comes down to bringing similar terms. After this, you should carefully read the condition again and think about how we can get rid of the monomial. It is obvious that for this you need to add the same monomial to it, but with the opposite sign - . Next, we replace the asterisk with this monomial and make sure that our solution is correct.

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

Additional materials
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Electronic textbook based on the textbook by Yu.N. Makarycheva
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Guys, you have already studied monomials in the topic: Standard form of a monomial. Definitions. Examples. Let's review the basic definitions.

Monomial– an expression consisting of a product of numbers and variables. Variables can be raised to natural powers. A monomial does not contain any operations other than multiplication.

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

Similar monomials– these are either identical monomials, or monomials that differ from each other by a coefficient.

The concept of a polynomial

Definition of a polynomial

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 there are three, then with a trinomial. If there are more terms, it is a polynomial.

Examples of polynomials.

1) 2аb + 4сd (binomial);

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

3) 4a 2 b 4 + 4c 8 d 9 + 2xу 3 ;

3с 7 d 8 - 2b 6 c 2 d + 7xу - 5xy 2.

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

But a 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 open the brackets, the resulting expression will be a polynomial.
3b + 2a * (c 2 + d) = 3b + 2ac 2 + 2ad.

Polynomial degree 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 polynomial

The polynomial is brought to a standard form in order to remove unnecessary cumbersome writing and 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 standard form, you need to:
1. bring all its members to a standard form,
2. add similar (identical or with different numerical coefficients) terms. This procedure is often called bringing similar.

Example.
Reduce the polynomial aba + 2y 2 x 4 x + y 2 x 3 x 2 + 4 + 10a 2 b + 10 to 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's determine the powers 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 has the zero degree.
This means that we will put 3 x 5 y 2 (7th degree) in first place, 12a 2 b (3rd degree) in second place, and 14 (zero degree) in third place.
As a result, we obtain a polynomial of the standard form 3x 5 y 2 + 11a 2 b + 14.

Examples for self-solution

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) 4ax 2 + 5bc - 6a - 24bc + xx 4 x (5ax 6 - 19bc - 6a);

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

The terms of a polynomial are the basic units of many algebraic structures. By definition, monomials are either natural numerical values ​​or certain variables (groups of variables multiplied by each other).

One of the main mathematical operations on a polynomial is the reduction of similar terms. In this video tutorial we will look in more detail at what operations on a polynomial are.

Since all terms of a polynomial are related to each other through algebraic summation, they are all called terms. Monomials that have the same letter part are similar, i.e. consisting of identical variables. In this case, the variables must be in the same degree and with an equal numerical coefficient. And individual numerical values ​​in polynomials are considered equivalent to similar terms in themselves.

Reducing similar terms involves grouping the monomials of a polynomial so that separate parts are obtained, consisting entirely of similar terms. For example, consider this polynomial:

3a 2 + 2ab 2 - 6 - 3c 3 + 6a 2 - 7ab 2 + 7

Similar terms, in this case, are:

1. All free numeric values: -6, +7;
2. Monomials with base a squared: +3a 2, +6a 2;
3. Monomials with base ab squared: 2ab 2, -7ab 2;
4. Monomials with base c cubed: -3c 3 ;

The last group consists of only one monomial, which has no similar one in the entire polynomial.

Why are such transformations needed? Bringing similar terms helps to simplify the polynomial, bringing it to an elementary form, which consists of fewer monomials. This can be easily done by grouping those terms between which algebraic operations are performed. The main operations here are subtraction and addition - they also have the effect of rearrangement and allow you to freely move monomials inside the polynomial. Therefore, it is quite according to the rules to transform the above example like this:

6 +7 + 3a 2 +6a 2 + 2ab 2 +(-7ab 2) + (-3c 3) =

9а 2 - 5ab 2 - 3с 3 - 1

By implementing standard subtraction and addition, we obtain a simplified polynomial. If the original version had 7 monomials, then the current one has only 4 members. However, a logical question arises: what is the exact criterion for the “simplicity” of a polynomial?
From the point of view of algebraic rules, an elementary, or more precisely, a standard polynomial is considered to be a polynomial in which all the bases of the monomials are different and are not similar to each other. Our example:

9а 2 - 5ab 2 - 3с 3 - 1

Consists of monomials with bases a 2, ab 2, c 3, as well as one numerical value. None of the above items can be added to or subtracted from another. Before us is a standard polynomial consisting of four terms.

Any polynomial has such a criterion as degree. The degree of a polynomial, in general terms, is the largest degree of a monomial in a given polynomial. It is worth learning an important detail - the degrees of multi-letter (multivariable) expressions are summed up. Therefore, the total power of ab 2 is three (a to the first power, b squared). A polynomial of the form:

9а 2 - 5ab 2 - 3с 3 - 1

has a degree equal to three, since one of the monomials is in the largest cubic power.

The degree of polynomials is usually determined only for the standard form. If a polynomial has similar terms, then it is first reduced to a simplified form, and then the final degree is calculated.

If a polynomial consists of only numerical monomials, then its standard form takes the form of a singular number, which is the algebraic sum of all monomials. The degree of a given number, as a polynomial, is zero. If the number itself, being a standard type of polynomial, acquires the value “zero,” then its degree is considered indefinite, and the “zero” polynomial itself is called a null polynomial.

In the presented video it is also noticeable that any polynomial has, among other things, a leading coefficient and a free term. The leading coefficient is the numerical value that stands in front of the variable with the highest degree (the one that specifies the rank of the polynomial itself). And the free term is the total sum of all numerical values ​​of the polynomial. If there are no similar values ​​in the polynomial, or if they cancel completely, then the free term is taken equal to 0. In the example:

7a 4 - 2b 2 + 5c 3 + 3

the highest coefficient is the number 7, because it comes before the variable that has the highest degree (the fourth - and, at the same time, the entire polynomial has the fourth degree). The free term, in this example, is 3.