What does it mean to indicate the degree of a polynomial. The meaning of the word polynomial

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 sometimes also called a polynomial, and monomials that are part of a polynomial are members of a polynomial or monomials.

The complex concept of a polynomial

If a polynomial consists of two terms, then it is called a binomial, if it consists of three - a trinomial. The names four-term, five-term 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 the point of view of mathematics, a monomial is a special case of a polynomial. A monomial is a polynomial that has only 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 monomials included in it as terms are written in standard form and similar terms are given.

Standard form of a polynomial

The procedure for bringing a polynomial to the standard form is to bring each of the monomials to the standard form, and then add all such monomials together. The addition of similar members of a polynomial is called reduction of similar terms.
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 the polynomial.

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

When the polynomial is reduced to the standard form, we can talk about such a concept as the degree of the polynomial. The degree of a polynomial is the largest 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 a monomial included in the polynomial (5*x^3*y^2) is the fifth.

The concept of a polynomial

Definition of a polynomial: A polynomial is the sum of monomials. Polynomial example:

here we see the sum of two monomials, and this is the polynomial, i.e. sum of monomials.

The terms that make up a polynomial are called members of the polynomial.

Is the difference of monomials a polynomial? Yes, it is, because the difference is easily reduced to the sum, for example: 5a - 2b = 5a + (-2b).

Monomials are also considered polynomials. But there is no sum in a monomial, then why is it considered a polynomial? And you can add zero to it and get its sum with a zero monomial. So, a monomial is a special case of a polynomial, it consists of one member.

The number zero is a zero polynomial.

Standard form of a polynomial

What is a standard form polynomial? A polynomial is the sum of monomials, and if all these monomials that make up a polynomial are written in standard form, in addition, there should be no similar ones among them, then the polynomial is written in standard form.

An example of a polynomial in standard form:

here the polynomial consists of 2 monomials, each of which has a standard form, among the monomials there are no similar ones.

Now an example of a polynomial that does not have a standard form:

here are two monomials: 2a and 4a are similar. We need to add them, then the polynomial will get a standard form:

Another example:

Is this polynomial reduced to standard form? No, its second member is not written in the standard form. Writing it in standard form, we obtain a standard form polynomial:

Degree of a polynomial

What is the degree of a polynomial?

Polynomial degree definition:

The degree of a polynomial is the largest degree that the monomials that make up a given polynomial of standard form have.

Example. What is the degree of the polynomial 5h? The degree of the polynomial 5h is equal to one, because this polynomial contains only one monomial and its degree is equal to one.

Another example. What is the degree of the polynomial 5a 2 h 3 s 4 +1? The degree of the polynomial 5a 2 h 3 s 4 + 1 is nine, because this polynomial includes two monomials, the first monomial 5a 2 h 3 s 4 has the highest degree, and its degree is 9.

Another example. What is the degree of polynomial 5? The degree of the polynomial 5 is zero. So, the degree of a polynomial consisting only of a number, i.e. without letters, is equal to zero.

Last example. What is the degree of the zero polynomial, i.e. zero? The degree of the zero polynomial is not defined.

After studying monomials, we turn to polynomials. This article will tell you about all the necessary information needed 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 a standard form, introduce a degree and learn how to find it, work with its coefficients.

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Polynomial and its members - definitions and examples

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

Definition 1

polynomial the sum of monomials is considered, and the monomial itself is a special case of a polynomial.

It follows from the definition 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

The members of the polynomial its constituent monomials are called.

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

Definition 3

Polynomials that have 2, 3 trinomials in their composition have the corresponding name - binomial and trinomial.

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

According to the school curriculum, they worked with a linear binomial of the form a x + b, where a and b are some numbers, and x is a variable. 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 .

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

Definition 4

Similar members of a polynomial are like terms in the 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.

Standard form polynomial

All monomials and polynomials have their own specific names.

Definition 5

Standard form polynomial A polynomial is called in which each member of it has a monomial of the standard form and does not contain similar members.

It can be seen from the definition that it is possible to reduce polynomials of standard form, for example, 3 x 2 − x y + 1 and __formula__, and the record 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 the standard form, since the first of them has similar terms in the form 3 x 2 and − x2, and the second contains a monomial of the form x · y 3 · x · z 2 , which differs from the standard polynomial.

If circumstances so require, 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 member of the polynomial is a standard form polynomial without a letter part.

In other words, when the notation of a polynomial in standard form has a number, it is called a free member. Then the number 5 is a free member of the polynomial x 2 · z + 5 , and the polynomial 7 · a + 4 · a · b + b 3 has no free member.

The degree of a polynomial - how to find it?

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

Definition 7

The degree of a standard form polynomial name the largest of the powers 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 largest of them is 3, respectively. The definition of the degree from the polynomial 4 x 2 y 3 − 5 x 4 y + 6 x equals the largest of the numbers, that is, 2 + 3 = 5 , 4 + 1 = 5 and 1 , so 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 the 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 a c b + y 2 z 2 − 2 a 12 − a 12.

Solution

First, we present the polynomial in the standard form. We get an expression like:

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

When obtaining a polynomial of the standard form, we find that two of them are clearly distinguished - 2 · a 2 · b 2 · c 2 and y 2 · z 2 . To find the degrees, we calculate and get that 2 + 2 + 2 = 6 and 2 + 2 = 4 . It can be seen that the largest of them is equal to 6. It follows from the definition that exactly 6 is the degree of the polynomial − 2 · a 2 · b 2 · c 2 + y 2 · z 2, hence the original value.

Answer: 6 .

The coefficients of the terms of the polynomial

Definition 9

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

When considering the example, it is clear that the polynomial of the form 2 x − 0, 5 x y + 3 x + 7 has 4 polynomials in its composition: 2 x, − 0, 5 x y, 3 x and 7 with their respective coefficients 2 , − 0 , 5 , 3 and 7 . Hence, 2 , − 0 , 5 , 3 and 7 are considered to be the coefficients of the terms of the 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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Or, strictly, a finite formal sum of the form

∑ I c I x 1 i 1 x 2 i 2 ⋯ x n i n (\displaystyle \sum _(I)c_(I)x_(1)^(i_(1))x_(2)^(i_(2))\ cdots x_(n)^(i_(n))), where

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

c 0 + c 1 x 1 + ⋯ + c m x ​​m (\displaystyle c_(0)+c_(1)x^(1)+\dots +c_(m)x^(m)), where

With the help of a polynomial, the concepts of "algebraic equation" and "algebraic function" are derived.

Study and application[ | ]

The study of polynomial equations and their solutions was almost the main object of "classical algebra".

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

The technical simplicity of computations involving 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 the Weierstrass approximation theorem), contributed to the development of series expansion methods and polynomial Interpolation in Calculus.

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

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

Related definitions[ | ]

• Kind polynomial 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 a 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, is called polynomial carrier, and its convex hull is Newton's polyhedron.
• The degree of the polynomial is the maximum of the powers of its monomials. The degree of identical zero is further defined 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 tripartite.
• The coefficients of a polynomial are usually taken from a certain commutative ring R (\displaystyle R)(most often fields, such as 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 polynomial p (x) (\displaystyle p(x)) one variable, solution of 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

p R: A → A (\displaystyle p_(R):A\to A).

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

If 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 determines the polynomial p. However, this is not true in general, 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 pq (\displaystyle pq) is divisible by an irreducible polynomial, then p or q divided by λ (\displaystyle \lambda ). Each polynomial of degree greater than zero decomposes in a given field into a product of irreducible factors in a unique way (up to factors of degree zero).

For example, polynomial x 4 − 2 (\displaystyle x^(4)-2), which is 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, every 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 main theorem of algebra).

For two or more variables, this can no longer be asserted. Over any field for any 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.