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

Polynomial and its terms - definitions and examples

The definition of a polynomial was given in 7 class after studying monomials. Let's look at its full definition.

Definition 1

Polynomial The sum of monomials is calculated, and the monomial itself is a special case of a polynomial.

From the definition it follows that examples of polynomials can 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.

Let's look at some more definitions.

Definition 2

Members of the polynomial its constituent monomials are called.

Consider an example where we have a polynomial 3 x 4 − 2 x y + 3 − y 3, consisting of 4 terms: 3 x 4, − 2 x y, 3 and − y 3. Such a monomial can be considered a polynomial, which consists of one term.

Definition 3

Polynomials that contain 2, 3 trinomials have the corresponding name - binomial And trinomial.

It follows that an expression of the form x+y– is a binomial, and the expression 2 x 3 q − q x x x + 7 b is a trinomial.

By school curriculum worked with a linear binomial of the form a · x + b, where a and b are some numbers, and x is a variable. Let's consider examples of linear binomials of the form: x + 1, x · 7, 2 − 4 with examples of square trinomials x 2 + 3 · x − 5 and 2 5 · x 2 - 3 x + 11.

To transform and solve, it is necessary to find and bring similar terms. For example, a polynomial of the form 1 + 5 x − 3 + y + 2 x has similar terms 1 and - 3, 5 x and 2 x. They are divided into a special group called similar members of the polynomial.

Definition 4

Similar terms of a polynomial are similar terms found in a polynomial.

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

Polynomial of standard form

All monomials and polynomials have their own specific names.

Definition 5

Polynomial of standard form is a polynomial in which each term included in it has a monomial of standard form and does not contain similar terms.

From the definition it is clear that it is possible to reduce polynomials of the standard form, for example, 3 x 2 − x y + 1 and __formula__, and the entry is in standard form. The expressions 5 + 3 · x 2 − x 2 + 2 · x · z and 5 + 3 · x 2 − x 2 + 2 · x · z are not polynomials of standard form, since the first of them has similar terms in the form 3 · x 2 and − x 2, and the second contains a monomial of the form x · y 3 · x · z 2, which differs from the standard polynomial.

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

Definition 6

Free term of a polynomial is a polynomial of standard form that does not have a literal part.

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

Degree of a polynomial - how to find it?

The definition of the degree of a polynomial itself is based on the definition of a standard form polynomial and on the degrees of the monomials that are its components.

Definition 7

Degree of a polynomial of standard form is called the largest of the degrees included in its notation.

Let's look at an example. The degree of the polynomial 5 x 3 − 4 is equal to 3, because the monomials included in its composition have degrees 3 and 0, and the larger of them is 3, respectively. The definition of the degree from the polynomial 4 x 2 y 3 − 5 x 4 y + 6 x is equal to the largest of the numbers, that is, 2 + 3 = 5, 4 + 1 = 5 and 1, which means 5.

It is necessary to find out how the degree itself is found.

Definition 8

Degree of a polynomial of an arbitrary number is the degree of the corresponding polynomial in standard form.

When a polynomial is not written in standard form, but you need to find its degree, you need to reduce it to the standard form, and then find the required degree.

Example 1

Find the degree of a polynomial 3 a 12 − 2 a b c c a c b + y 2 z 2 − 2 a 12 − a 12.

Solution

First, let's present the polynomial in standard form. We get an expression of the form:

3 a 12 − 2 a b c c a c b + y 2 z 2 − 2 a 12 − a 12 = = (3 a 12 − 2 a 12 − a 12) − 2 · (a · a) · (b · b) · (c · c) + y 2 · z 2 = = − 2 · a 2 · b 2 · c 2 + y 2 · z 2

When obtaining a polynomial of standard form, we find that two of them stand out clearly - 2 · a 2 · b 2 · c 2 and y 2 · z 2 . To find the degrees, we count and find that 2 + 2 + 2 = 6 and 2 + 2 = 4. It can be seen that the largest of them is 6. From the definition it follows that 6 is the degree of the polynomial − 2 · a 2 · b 2 · c 2 + y 2 · z 2 , and therefore the original value.

Answer: 6 .

Coefficients of polynomial terms

Definition 9

When all terms of a polynomial are monomials of the standard form, then in this case they have the name coefficients of polynomial terms. In other words, they can be called coefficients of the polynomial.

When considering the example, it is clear that a polynomial of the form 2 x − 0, 5 x y + 3 x + 7 contains 4 polynomials: 2 x, − 0, 5 x y, 3 x and 7 with their corresponding coefficients 2, − 0, 5, 3 and 7. This means that 2, − 0, 5, 3 and 7 are considered coefficients of terms 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 in front of the variables.

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Definition 3.3. Monomial is an expression that is a product of numbers, variables and powers with a natural exponent.

For example, each of the expressions,
,
is a monomial.

They say that the monomial has standard view , if it contains only one numerical factor in the first place, and each product of identical variables in it is represented by a degree. The numerical factor of a monomial written in standard form is called coefficient of the monomial . By the power of the monomial is called the sum of the exponents of all its variables.

Definition 3.4. Polynomial called the sum of monomials. The monomials from which a polynomial is composed are calledmembers of the polynomial .

Similar terms - monomials in a polynomial - are called similar terms of the polynomial .

Definition 3.5. Polynomial of standard form called a polynomial in which all terms are written in standard form and similar terms are given.Degree of a polynomial of standard form is called the greatest of the powers of the monomials included in it.

For example, is a polynomial of standard form of the fourth degree.

Actions on monomials and polynomials

The sum and difference of polynomials can be converted into a polynomial of standard form. When adding two polynomials, all their terms are written down and similar terms are given. When subtracting, the signs of all terms of the polynomial being subtracted are reversed.

For example:

The terms of a polynomial can be divided into groups and enclosed in parentheses. Since this is an identical transformation inverse to the opening of parentheses, the following is established bracketing rule: if a plus sign is placed before the brackets, then all terms enclosed in brackets are written with their signs; If a minus sign is placed before the brackets, then all terms enclosed in brackets are written with opposite signs.

For example,

Rule for multiplying a polynomial by a polynomial: To multiply a polynomial by a polynomial, it is enough to multiply each term of one polynomial by each term of another polynomial and add the resulting products.

For example,

Definition 3.6. Polynomial in one variable degrees called an expression of the form

Where
- any numbers that are called polynomial coefficients , and
,– non-negative integer.

If
, then the coefficient called leading coefficient of the polynomial
, monomial
- his senior member , coefficient – free member .

If instead of a variable to a polynomial
substitute real number , then the result will be a real number
which is called the value of the polynomial
at
.

Definition 3.7. Number calledroot of the polynomial
, If
.

Consider dividing a polynomial by a polynomial, where
And - integers. Division is possible if the degree of the polynomial dividend is
not less than the degree of the divisor polynomial
, that is
.

Divide a polynomial
to a polynomial
,
, means finding two such polynomials
And
, to

In this case, the polynomial
degrees
called polynomial-quotient ,
– the remainder ,
.

Remark 3.2. If the divisor
–is not a zero polynomial, then division
on
,
, is always feasible, and the quotient and remainder are uniquely determined.

Remark 3.3. In case
in front of everyone , that is

they say that it is a polynomial
completely divided(or shares)to a polynomial
.

The division of polynomials is carried out similarly to the division of multi-digit numbers: first, the leading term of the dividend polynomial is divided by the leading term of the divisor polynomial, then the quotient from the division of these terms, which will be the leading term of the quotient polynomial, is multiplied by the divisor polynomial and the resulting product is subtracted from the dividend polynomial . As a result, a polynomial is obtained - the first remainder, which is divided by the divisor polynomial in a similar way and the second term of the quotient polynomial is found. This process is continued until a zero remainder is obtained or the degree of the remainder polynomial is less than the degree of the divisor polynomial.

When dividing a polynomial by a binomial, you can use Horner's scheme.

Horner scheme

Suppose we want to divide a polynomial

by binomial
. Let us denote the quotient of division as a polynomial

and the remainder - . Meaning , polynomial coefficients
,
and the remainder Let's write it in the following form:

In this scheme, each of the coefficients
,
,
, …,obtained from the previous number in the bottom line by multiplying by the number and adding to the resulting result the corresponding number in the top line above the desired coefficient. If any degree is absent in the polynomial, then the corresponding coefficient is zero. Having determined the coefficients according to the given scheme, we write the quotient

and the result of division if
,

or ,

If
,

Theorem 3.1. In order for an irreducible fraction (

,

)was the root of the polynomial
with integer coefficients, it is necessary that the number was a divisor of the free term , and the number - divisor of the leading coefficient .

Theorem 3.2. (Bezout's theorem ) Remainder from dividing a polynomial
by binomial
equal to the value of the polynomial
at
, that is
.

When dividing a polynomial
by binomial
we have equality

This is true, in particular, when
, that is
.

Example 3.2. Divide by
.

Solution. Let's apply Horner's scheme:

Hence,

Example 3.3. Divide by
.

Solution. Let's apply Horner's scheme:

Hence,

,

Example 3.4. Divide by
.

Solution.

As a result we get

Example 3.5. Divide
on
.

Solution. Let's divide the polynomials by column:

Then we get

.

Sometimes it is useful to represent a polynomial as an equal product of two or more polynomials. Such an identity transformation is called factoring a polynomial . Let us consider the main methods of such decomposition.

Taking the common factor out of brackets. In order to factor a polynomial by taking the common factor out of brackets, you must:

1) find the common factor. To do this, if all the coefficients of the polynomial are integers, the largest modulo common divisor of all coefficients of the polynomial is considered as the coefficient of the common factor, and each variable included in all terms of the polynomial is taken with the largest exponent it has in this polynomial;

2) find the quotient of dividing a given polynomial by a common factor;

3) write down the product of the general factor and the resulting quotient.

Grouping of members. When factoring a polynomial using the grouping method, its terms are divided into two or more groups so that each of them can be converted into a product, and the resulting products would have a common factor. After this, the method of bracketing the common factor of the newly transformed terms is used.

Application of abbreviated multiplication formulas. In cases where the polynomial to be expanded into factors, has the form of the right side of any abbreviated multiplication formula; its factorization is achieved by using the corresponding formula written in a different order.

Let

, then the following are true abbreviated multiplication formulas:

For :
If odd ( ):
Newton binomial: Where – number of combinations of By .

Introduction of new auxiliary members. This method consists in replacing a polynomial with another polynomial that is identically equal to it, but containing a different number of terms, by introducing two opposite terms or replacing any term with an identically equal sum of similar monomials. The replacement is made in such a way that the method of grouping terms can be applied to the resulting polynomial.

Example 3.6..

Solution. All terms of a polynomial contain a common factor
. Hence,.

Answer: .

Example 3.7.

Solution. We group separately the terms containing the coefficient , and terms containing . Taking the common factors of groups out of brackets, we get:

.

Answer:
.

Example 3.8. Factor a polynomial
.

Solution. Using the appropriate abbreviated multiplication formula, we get:

Answer: .

Example 3.9. Factor a polynomial
.

Solution. Using the grouping method and the corresponding abbreviated multiplication formula, we obtain:

.

Answer: .

Example 3.10. Factor a polynomial
.

Solution. We will replace on
, group the terms, apply the abbreviated multiplication formulas:

.

Answer:
.

Example 3.11. Factor a polynomial

Solution. Because ,
,
, That

- polynomials. In this article we will outline all the initial and necessary information about polynomials. These include, firstly, the definition of a polynomial with accompanying definitions of the terms of the polynomial, in particular, the free term and similar terms. Secondly, we will dwell on polynomials of the standard form, give the corresponding definition and give examples of them. Finally, we will introduce the definition of the degree of a polynomial, figure out how to find it, and talk about the coefficients of the terms of the polynomial.

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Polynomial and its terms - definitions and examples

In grade 7, polynomials are studied immediately after monomials, this is understandable, since polynomial definition is given through monomials. Let us give this definition to explain what a polynomial is.

Definition.

Polynomial is the sum of monomials; A monomial is considered a special case of a polynomial.

The written definition allows you to give as many examples of polynomials as you like. Any of the monomials 5, 0, −1, x, 5 a b 3, x 2 0.6 x (−2) y 12, etc. is a polynomial. Also, by definition, 1+x, a 2 +b 2 and are polynomials.

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

Definition.

Polynomial terms are the constituent monomials of a polynomial.

For example, the polynomial 3 x 4 −2 x y+3−y 3 consists of four terms: 3 x 4 , −2 x y , 3 and −y 3 . A monomial is considered a polynomial consisting of one term.

Definition.

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

So x+y is a binomial, and 2 x 3 q−q x x x+7 b is a trinomial.

At school, we most often have to work with linear binomial a x+b , where a and b are some numbers, and x is a variable, as well as c quadratic trinomial a·x 2 +b·x+c, where a, b and c are some numbers, and x is a variable. Here are examples of linear binomials: x+1, x 7,2−4, and here are examples of square trinomials: x 2 +3 x−5 and .

Polynomials in their notation can have similar terms. For example, in the polynomial 1+5 x−3+y+2 x the similar terms are 1 and −3, as well as 5 x and 2 x. They have their own special name - similar terms of a polynomial.

Definition.

Similar terms of a polynomial similar terms in a polynomial are called.

In the previous example, 1 and −3, as well as the pair 5 x and 2 x, are similar terms of the polynomial. In polynomials that have similar terms, you can reduce similar terms to simplify their form.

Polynomial of standard form

For polynomials, as for monomials, there is a so-called standard form. Let us voice the corresponding definition.

Based this definition, we can give examples of polynomials of standard form. So the polynomials 3 x 2 −x y+1 and written in standard form. And the expressions 5+3 x 2 −x 2 +2 x z and x+x y 3 x z 2 +3 z are not polynomials of the standard form, since the first of them contains similar terms 3 x 2 and −x 2 , and in the second – a monomial x·y 3 ·x·z 2 , the form of which is different from the standard one.

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

Another concept related to polynomials of the standard form is the concept of a free term of a polynomial.

Definition.

Free term of a polynomial is a member of a polynomial of standard form without a letter part.

In other words, if a polynomial of standard form contains a number, then it is called a free member. For example, 5 is the free term of the polynomial x 2 z+5, but the polynomial 7 a+4 a b+b 3 does not have a free term.

Degree of a polynomial - how to find it?

Another important related definition is the definition of the degree of a polynomial. First, we define the degree of a polynomial of the standard form; this definition is based on the degrees of the monomials that are in its composition.

Definition.

Degree of a polynomial of standard form is the largest of the powers of the monomials included in its notation.

Let's give examples. The degree of the polynomial 5 x 3 −4 is equal to 3, since the monomials 5 x 3 and −4 included in it have degrees 3 and 0, respectively, the largest of these numbers is 3, which is the degree of the polynomial by definition. And the degree of the polynomial 4 x 2 y 3 −5 x 4 y+6 x equal to the largest of the numbers 2+3=5, 4+1=5 and 1, that is, 5.

Now let's find out how to find the degree of a polynomial of any form.

Definition.

The degree of a polynomial of arbitrary form call the degree of the corresponding polynomial of standard form.

So, if a polynomial is not written in standard form, and you need to find its degree, then you need to reduce the original polynomial to standard form, and find the degree of the resulting polynomial - it will be the required one. Let's look at the example solution.

Example.

Find the degree of the polynomial 3 a 12 −2 a b c a c b+y 2 z 2 −2 a 12 −a 12.

Solution.

First you need to represent the polynomial in standard form:
3 a 12 −2 a b c a c b+y 2 z 2 −2 a 12 −a 12 = =(3 a 12 −2 a 12 −a 12)− 2·(a·a)·(b·b)·(c·c)+y 2 ·z 2 = =−2 a 2 b 2 c 2 +y 2 z 2.

The resulting polynomial of standard form includes two monomials −2·a 2 ·b 2 ·c 2 and y 2 ·z 2 . Let's find their powers: 2+2+2=6 and 2+2=4. Obviously, the largest of these powers is 6, which by definition is the power of a polynomial of the standard form −2 a 2 b 2 c 2 +y 2 z 2, and therefore the degree of the original polynomial., 3 x and 7 of the polynomial 2 x−0.5 x y+3 x+7 .

Bibliography.

• 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. At 2 p.m. Part 1. Textbook for students educational 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.

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 the polynomial:

7a + 2b - 3c - 11

expressions: 7 a, 2b, -3c and -11 are the terms of the polynomial. Notice the -11 term. It does not contain a variable. Such members consisting only of numbers are called free.

It is generally accepted that any monomial is a special case of a 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 terms, there are also special names - binomial and trinomial, respectively:

7a- monomial

7a + 2b- binomial

7a + 2b - 3c- trinomial

Similar members

Similar members- monomials included in a polynomial that differ from each other only by 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 members 5 abc 2 and -7 abc 2 are similar terms.

Bringing similar members

If a polynomial contains similar terms, then it can be reduced to a simpler form by combining similar terms into one. This action is called bringing similar members. First of all, let’s put all such terms separately in brackets:

(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

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

Polynomial of standard form

Polynomial of standard form is a polynomial all of whose terms are monomials of standard form, among which there are no similar terms.

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

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

First, let's find similar terms:

If all members of a standard form polynomial contain the same variable, then its members are usually arranged from greatest to least degree. The free term of the polynomial, if there is one, is placed in last place - on the right.

For example, a polynomial

3x + x 3 - 2x 2 - 7

should be written like this:

x 3 - 2x 2 + 3x - 7

It is strange that an equality is made between a polynomial and a polynomial. Although as far as I remember these are different things. A polynomial is what they write about here. A polynomial is the ratio of 2 polynomials. I looked up the translation in the dictionary english words I saw a polynomial that was translated as polynomial and I was quite surprised... It turns out they don't even see the difference. Regarding the 1st example... This is all good, but is there a way to directly convert without entering unknown coefficients? This method is too pretentious... There is a lot to be said about polynomials. This goes far beyond the scope of school. Research is still ongoing! Those. The topic of polynomials is not completed. I can answer the question about roots in radicals. In general, it has been proven that polynomials of degree above 4 do not have solutions in radicals. And they cannot be solved analytically at all. Although some types are quite solvable. But not all... The 3rd degree equation has a Cardano solution. A 4th degree equation has 2 types of formulas. They are quite complex and in general it is not clear in advance whether there are valid solutions; they can all be complex. A polynomial of odd degree always has at least 1 real root. In theory, formulas for solving equations of even the 3rd or 4th degree are not particularly widespread due to their complexity. And the question arises as to which roots to consider. After all, an equation of nth degree has exactly n roots, taking into account their multiplicity. For example, you can solve an equation numerically using Newton’s method. Everything is simple there. An iteration formula is written and there are no problems. Linear approximation. The straight line intersects with the OX axis only at the 1st point. May not intersect, then the root is complex. But also 1st. Well, it’s clear that if a polynomial with real coefficients has complex root, then it also has a complex conjugate. However, already in the quadratic approximation (this method is called the parabola method and other variants of this Muller method based on the 2 previous points, etc.) problems arise. Firstly, there are 2 roots (MB if the discriminant > 0) which one to choose? Although the equation is quadratic. You can go further, take the cubic approximation (the 4th term in the Taylor series, for q we take 3) and even the 4th degree approximation by taking 5 terms of the Taylor series. Convergence will be super fast. Everything can be solved analytically! But I have never seen such methods anywhere in the mathematical literature. As a rule, they use Newton's method because it is problem-free! And wherever cubic or fourth-degree equations occur in theory, this happens. If you want, try it yourself! I don't think you will be delighted. Although I repeat, everything is solved analytically. The formulas will just be very complicated. But that's not the point. Lots of other problems arise that are not related to complexity.