Or, strictly, is 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))), WhereIn 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)), WhereUsing 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 whole series 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 the 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
p R: A → A (\displaystyle p_(R):A\to A).The most frequently considered case is 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 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.
The concept of a polynomial
Definition of polynomial: A polynomial is the sum of monomials. Polynomial example:
here we see the sum of two monomials, and this is a polynomial, i.e. sum of monomials.
The terms that make up a polynomial are called terms of the polynomial.
Is the difference of monomials a polynomial? Yes, it is, because the difference is easily reduced to a sum, example: 5a – 2b = 5a + (-2b).
Monomials are also considered polynomials. But a monomial has no sum, 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 term.
The number zero is the zero polynomial.
Standard form of polynomial
What is a polynomial of standard form? A polynomial is the sum of monomials, and if all these monomials that make up the polynomial are written in standard form, and 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 two monomials: 2a and 4a are similar. You need to add them up, then the polynomial will take the standard form:
Another example:
Is this polynomial reduced to standard form? No, his second term is not written in standard form. Writing it in standard form, we obtain a polynomial of standard form:
Polynomial degree
What is the degree of a polynomial?
Polynomial degree definition:
The degree of a polynomial is the highest 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 equal to 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 the polynomial 5? The degree of a polynomial 5 is zero. So, the degree of a polynomial consisting only of a number, i.e. without letters, equals zero.
The last example. What is the degree of the zero polynomial, i.e. zero? The degree of the zero polynomial is not defined.
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 ).
With regard to the coefficients of mathematics, 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 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 axis ). 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.
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.
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Polynomial and its terms - definitions and examples
The definition of a polynomial was necessary back 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.
According to the school curriculum, we 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 a 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 9When all the 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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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 present 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 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.