Polynomials degree and standard form of polynomial. Meaning of the word polynomial

- 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, let us 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 on this definition, we can give examples of polynomials of the 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 .

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.

polynomial, expression of the form

Axkyl┘..wm + Bxnyp┘..wq + ┘┘ + Dxrts┘..wt,

where x, y, ..., w ≈ variables, and A, B, ..., D (M coefficients) and k, l, ..., t (exponents ≈ non-negative integers) ≈ constants. Individual terms of the form Ахkyl┘..wm are called terms of M. The order of the terms, as well as the order of the factors in each term, can be changed arbitrarily; in the same way, you can introduce or omit terms with zero coefficients, and in each individual term ≈ powers with zero coefficients. When a structure has one, two, or three members, it is called a monomial, binomial, or trinomial. Two terms of a equation are called similar if their exponents for identical variables are pairwise equal. Similar members

A"хkyl┘..wm, B"xkyl┘..wm, ┘.., D"xkyl┘..wm

can be replaced by one (bringing similar terms). Two models are called equal if, after reducing similar ones, all terms with non-zero coefficients turn out to be pairwise identical (but perhaps written in a different order), and also if all the coefficients of these models turn out to be equal to zero. In the latter case, the quantity is called identical zero and is denoted by the sign 0. The quantity of one variable x can always be written in the form

P(x) = a0xn+ a1xn-1 + ... + an-1x+ an,

where a0, a1,..., an ≈ coefficients.

The sum of the exponents of any member of a model is called the degree of that member. If M is not identically zero, then among the terms with nonzero coefficients (it is assumed that all such terms are given) there are one or more of the highest degree; this greatest degree is called the degree of M. The identical zero has no degree. M. of zero degree is reduced to one term A (constant, not equal to zero). Examples: xyz + x + y + z is a polynomial of the third degree, 2x + y ≈ z + 1 is a polynomial of the first degree (linear M), 5x2 ≈ 2x2 ≈ 3x2 has no degree, since it is identically zero. A model, all of whose members are of the same degree, is called a homogeneous model, or form; forms of the first, second and third degrees are called linear, quadratic, cubic, and according to the number of variables (two, three) binary (binary), trigeminal (ternary) (for example, x2 + y2 + z2 ≈ xy ≈ yz ≈ xz is a trigeminal quadratic form ).

Regarding the coefficients of algebra, it is assumed that they belong to a certain field (see Algebraic field), for example, the field of rational, real, or complex numbers. By performing the operations of addition, subtraction, and multiplication on a model based on the commutative, combinational, and distributive laws, one again obtains a model. Thus, the set of all models with coefficients from a given field forms a ring (see Algebraic ring) ≈ a ring of polynomials over a given field; this ring has no zero divisors, that is, the product of numbers not equal to 0 cannot give 0.

If for two polynomials P(x) and Q(x) it is possible to find a polynomial R(x) such that P = QR, then P is said to be divisible by Q; Q is called a divisor, and R ≈ quotient. If P is not divisible by Q, then one can find polynomials P(x) and S(x) such that P = QR + S, and the degree of S(x) is less than the degree of Q(x).

By repeatedly applying this operation, one can find the greatest common divisor of P and Q, that is, a divisor of P and Q that is divisible by any common divisor of these polynomials (see Euclidean algorithm). A matrix that can be represented as a product of a matrix of lower degrees with coefficients from a given field is called reducible (in a given field), otherwise it is called irreducible. Irreducible numbers play a role in the ring of numbers similar to that of prime numbers in the theory of integers. So, for example, the theorem is true: if the product PQ is divisible by an irreducible polynomial R, but P is not divisible by R, then Q must be divisible by R. Every M of degree greater than zero can be decomposed in a given field into a product of irreducible factors in a unique way ( up to zero degree factors). For example, the polynomial x4 + 1, irreducible in the field of rational numbers, is factorized

in the field of real numbers and by four factors ═in the field of complex numbers. In general, every model of one variable x is decomposed in the field of real numbers into factors of the first and second degree, and in the field of complex numbers into factors of the first degree (the fundamental theorem of algebra). For two or more variables this can no longer be said; for example, the polynomial x3 + yz2 + z3 is irreducible in any number field.

If the variables x, y, ..., w are given certain numerical values ​​(for example, real or complex), then M will also receive a certain numerical value. It follows that each model can be considered as a function of the corresponding variables. This function is continuous and differentiable for any values ​​of the variables; it can be characterized as an entire rational function, that is, a function obtained from variables and some constants (coefficients) through addition, subtraction and multiplication performed in a certain order. Entire rational functions are included in a broader class of rational functions, where division is added to the listed actions: any rational function can be represented as a quotient of two M. Finally, rational functions are contained in the class of algebraic functions.

One of the most important properties of mathematics is that any continuous function can be replaced with an arbitrarily small error by mathematics (Weierstrass’s theorem; its exact formulation requires that the given function be continuous on some limited, closed set of points, for example, on a segment of the real line ). This fact, proven by means of mathematical analysis, makes it possible to approximately express mathematically any relationship between quantities studied in any issue of natural science and technology. Methods for such an expression are studied in special sections of mathematics (see Approximation and interpolation of functions, Least squares method).

In elementary algebra, a polynomial is sometimes called an algebraic expression in which the last action is addition or subtraction, for example

Lit. : Kurosh A.G., Course of Higher Algebra, 9th ed., M., 1968; Mishina A.P., Proskuryakov I.V., Higher algebra, 2nd ed., M., 1965.

By definition, a polynomial is an algebraic expression representing the sum of monomials.

For example: 2*a^2 + 4*a*x^7 - 3*a*b^3 + 4; 6 + 4*b^3 are polynomials, and the expression z/(x - x*y^2 + 4) is not a polynomial because it is not a sum of monomials. A polynomial is also sometimes called a polynomial, and monomials that are part of a polynomial are members of a polynomial or monomials.

Complex concept of polynomial

If a polynomial consists of two terms, then it is called a binomial; if it consists of three, it is called a trinomial. The names fournomial, fivenomial and others are not used, and in such cases they simply say polynomial. Such names, depending on the number of terms, put everything in its place.

And the term monomial becomes intuitive. From a mathematical point of view, a monomial is a special case of a polynomial. A monomial is a polynomial that consists of one term.

Just like a monomial, a polynomial has its own standard form. The standard form of a polynomial is such a notation of a polynomial in which all the monomials included in it as terms are written in a standard form and similar terms are given.

Standard form of polynomial

The procedure for reducing a polynomial to standard form is to reduce each of the monomials to standard form, and then add all similar monomials together. The addition of similar terms of a polynomial is called reduction of similar.
For example, let's give similar terms in the polynomial 4*a*b^2*c^3 + 6*a*b^2*c^3 - a*b.

The terms 4*a*b^2*c^3 and 6*a*b^2*c^3 are similar here. The sum of these terms will be the monomial 10*a*b^2*c^3. Therefore, the original polynomial 4*a*b^2*c^3 + 6*a*b^2*c^3 - a*b can be rewritten as 10*a*b^2*c^3 - a*b . This entry will be the standard form of a polynomial.

From the fact that any monomial can be reduced to a standard form, it also follows that any polynomial can be reduced to a standard form.

When a polynomial is reduced to a standard form, we can talk about such a concept as the degree of a polynomial. The degree of a polynomial is the highest degree of a monomial included in a given polynomial.
So, for example, 1 + 4*x^3 - 5*x^3*y^2 is a polynomial of the fifth degree, since the maximum degree of the monomial included in the polynomial (5*x^3*y^2) is fifth.

Or, strictly, is a finite formal sum of the form

In particular, a polynomial in one variable is a finite formal sum of the form

Using a polynomial, the concepts of “algebraic equation” and “algebraic function” are derived.

Study and Application[ | ]

The study of polynomial equations and their solutions was perhaps the main object of “classical algebra.”

A number of transformations in mathematics are associated with the study of polynomials: the introduction into the consideration of zero, negative, and then complex numbers, as well as the emergence of group theory as a branch of mathematics and the identification of classes of special functions in analysis.

The technical simplicity of calculations associated with polynomials compared to more complex classes of functions, as well as the fact that the set of polynomials is dense in the space of continuous functions on compact subsets of Euclidean space (see Weierstrass's approximation theorem), contributed to the development of series expansion and polynomial expansion methods. interpolation in mathematical analysis.

Polynomials also play a key role in algebraic geometry, whose object is sets defined as solutions to systems of polynomials.

The special properties of transforming coefficients when multiplying polynomials are used in algebraic geometry, algebra, knot theory, and other branches of mathematics to encode or express properties of various objects in polynomials.

Related definitions[ | ]

• Polynomial of the form c x 1 i 1 x 2 i 2 ⋯ x n i n (\displaystyle cx_(1)^(i_(1))x_(2)^(i_(2))\cdots x_(n)^(i_(n))) called monomial or monomial multi-index I = (i 1 , … , i n) (\displaystyle I=(i_(1),\dots ,\,i_(n))).
• Monomial corresponding to multi-index I = (0 , … , 0) (\displaystyle I=(0,\dots ,\,0)) called free member.
• Full degree(non-zero) monomial c I x 1 i 1 x 2 i 2 ⋯ x n i n (\displaystyle c_(I)x_(1)^(i_(1))x_(2)^(i_(2))\cdots x_(n)^(i_ (n))) called an integer | I | = i 1 + i 2 + ⋯ + i n (\displaystyle |I|=i_(1)+i_(2)+\dots +i_(n)).
• Many multi-indexes I, for which the coefficients c I (\displaystyle c_(I)) non-zero, called carrier of the polynomial, and its convex hull is Newton's polyhedron.
• Polynomial degree is called the maximum of the powers of its monomials. The degree of identical zero is further determined by the value − ∞ (\displaystyle -\infty ).
• A polynomial that is the sum of two monomials is called binomial or binomial,
• A polynomial that is the sum of three monomials is called trinomial.
• The coefficients of the polynomial are usually taken from a specific commutative ring R (\displaystyle R)(most often fields, for example, fields of real or complex numbers). In this case, with respect to the operations of addition and multiplication, the polynomials form a ring (moreover, an associative-commutative algebra over the ring R (\displaystyle R) without zero divisors) which is denoted R [ x 1 , x 2 , … , x n ] . (\displaystyle R.)
• For a polynomial p (x) (\displaystyle p(x)) one variable, solving the equation p (x) = 0 (\displaystyle p(x)=0) is called its root.

Polynomial functions[ | ]

Let A (\displaystyle A) there is an algebra over a ring R (\displaystyle R). Arbitrary polynomial p (x) ∈ R [ x 1 , x 2 , … , x n ] (\displaystyle p(x)\in R) defines a polynomial function

The most frequently considered case is A = R (\displaystyle A=R).

In case R (\displaystyle R) is a field of real or complex numbers (as well as any other field with an infinite number of elements), the function f p: R n → R (\displaystyle f_(p):R^(n)\to R) completely defines the polynomial p. However, in general this is not true, for example: polynomials p 1 (x) ≡ x (\displaystyle p_(1)(x)\equiv x) And p 2 (x) ≡ x 2 (\displaystyle p_(2)(x)\equiv x^(2)) from Z 2 [ x ] (\displaystyle \mathbb (Z)_(2)[x]) define identically equal functions Z 2 → Z 2 (\displaystyle \mathbb (Z) _(2)\to \mathbb (Z) _(2)).

A polynomial function of one real variable is called an entire rational function.

Types of polynomials[ | ]

Properties [ | ]

Divisibility [ | ]

The role of irreducible polynomials in the polynomial ring is similar to the role of prime numbers in the ring of integers. For example, the theorem is true: if the product of polynomials p q (\displaystyle pq) is divisible by an irreducible polynomial, then p or q divided by λ (\displaystyle \lambda). Each polynomial of degree greater than zero can be decomposed in a given field into a product of irreducible factors in a unique way (up to factors of degree zero).

For example, a polynomial x 4 − 2 (\displaystyle x^(4)-2), irreducible in the field of rational numbers, decomposes into three factors in the field of real numbers and into four factors in the field of complex numbers.

In general, each polynomial in one variable x (\displaystyle x) decomposes in the field of real numbers into factors of the first and second degree, in the field of complex numbers into factors of the first degree (the fundamental theorem of algebra).

For two or more variables this can no longer be said. Above any field for anyone n > 2 (\displaystyle n>2) there are polynomials from n (\displaystyle n) variables that are irreducible in any extension of this field. Such polynomials are called absolutely irreducible.