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Many authors use these two words interchangeably. A polynomial P in the indeterminate x is commonly denoted either as P or as P(x). Formally, the name of the polynomial is P, not P(x), but the use of the functional notation P(x) dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the ...
For example the above polynomial expression is equivalent (denote the same polynomial as + + Many author do not distinguish polynomials and polynomial expressions. In this case the expression of a polynomial expression as a linear combination is called the canonical form , normal form , or expanded form of the polynomial.
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. For example, in the polynomial + +, with variables and , the first two terms have the coefficients 7 and −3. The third term 1.5 is the constant coefficient.
The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution.
For example, polynomials are conventionally written with the terms in descending powers: it is more usual to write x 2 + x + 30 than x + 30 + x 2, although the two forms define the same polynomial. By contrast, the existence of Jordan canonical form for a matrix is a deep theorem.
The letter may be followed by a subscript: a number (as in x 2), another variable (x i), a word or abbreviation of a word as a label (x total) or a mathematical expression (x 2i+1). Under the influence of computer science , some variable names in pure mathematics consist of several letters and digits.
In abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any field. In this setting, given a field F and some indeterminate X , a rational expression (also known as a rational fraction or, in algebraic geometry , a rational function ) is any element ...
[2] [3] The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. [4]