Polynomial expression

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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 constants, 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.

  • 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 theorem 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.

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