Polynomial expression
Encyclopedia
In mathematics
, and in particular in the field of algebra
, a polynomial expression in one or more given entities E1, E2, ..., is any meaningful expression constructed from copies of those entities together with constant
s, using the operations of addition and multiplication. For each entity E, multiple copies can be used, and it is customary to write the product E×E×...×E of some number n of identical copies of E as En; thus the operation of raising to a constant natural number
power may also be used (as abbreviation) in a polynomial expression. Similarly, subtraction X – Y may be used to abbreviate X + (–1)×Y.
The entities used may be of various natures. They are usually not explicitly given values, since then the polynomial expression can just be evaluated to another such value. Often they are symbols such as "x", "λ" or "X", which according to the context may stand for an unknown quantity, a mathematical variable
, a parameter
, or an indeterminate
, and in such cases the polynomial expression is just a polynomial
. It is however also possible to form polynomial expressions in more complicated entities than just symbols. Here are examples of such uses of polynomial expressions.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and in particular in the field of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
, a polynomial expression in one or more given entities E1, E2, ..., is any meaningful expression constructed from copies of those entities together with constant
Constant (mathematics)
In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...
s, using the operations of addition and multiplication. For each entity E, multiple copies can be used, and it is customary to write the product E×E×...×E of some number n of identical copies of E as En; thus the operation of raising to a constant natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
power may also be used (as abbreviation) in a polynomial expression. Similarly, subtraction X – Y may be used to abbreviate X + (–1)×Y.
The entities used may be of various natures. They are usually not explicitly given values, since then the polynomial expression can just be evaluated to another such value. Often they are symbols such as "x", "λ" or "X", which according to the context may stand for an unknown quantity, a mathematical variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
, a parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
, or an indeterminate
Indeterminate
Indeterminate has a variety of meanings in mathematics:* Indeterminate * Indeterminate system* Indeterminate equation* Statically indeterminate* Indeterminate formIt is also a term in botany and gardening:*Indeterminate growth...
, and in such cases the polynomial expression is just a polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
. It is however also possible to form polynomial expressions in more complicated entities than just symbols. Here are examples of such uses of polynomial expressions.
- The entities may be themselves expressions, not necessarily polynomial ones. For instance, it is possible to use the de Moivre's identity for any integer n to express cos(nx) as a polynomial expression in (the entity) cos(x), as in cos(3x) = 4 cos(x)3 − 3 cos(x). Here it would be incorrect to call the right hand side a polynomial.
- The entities may be matrices; for instance the Cayley–Hamilton theoremCayley–Hamilton theoremIn linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation....
applied to a matrix A equates a certain polynomial expression in A to the null matrix. - The entries may be "somewhat unknown" quantities without being completely free variables. For instance, for any monic polynomial of degree n that has n roots, Viète's formulas express its coefficients as (symmetric) polynomial expressions in those roots. This means that the relations expressed by those formulas exist independently of the choice of such a polynomial; therefore the n roots are not known values (as they would be if the polynomial were fixed), but they are not variables or indeterminates either.