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Polynomial
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Everything about Polynomial totally explainedIn mathematics, a polynomial is an expression constructed from one or more variables and constants, using the operations of addition, subtraction, multiplication, and constant positive whole number exponents. For example, is a polynomial, but |
The names for degrees higher than are less common. The names for the degrees may be applied to the polynomial or to its terms. For example, a constant may refer to a zero degree polynomial or to a zero degree term.
The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree isn't zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined to be negative (either –1 or –∞)(External Link ). The latter convention is important when defining Euclidean division of polynomials.
| Number of non-zero terms |
Name |
Example |
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zero polynomial |
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monomial |
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binomial |
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trinomial |
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The word monomial can be ambiguous, as it's also often used to denote just a power of the variable, or in the multivariate case product of such powers, without any coefficient. Two or more terms which involve the same monomial in the latter sense, in other words which differ only in the value of their coefficients, are called similar terms; they can be combined into a single term by adding their coefficients; if the resulting term has coefficient zero, it may be removed altogether. The above classification according to the number of terms assumes that similar terms have been combined first.
Extensions of the concept of a polynomial
One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (R[X])[Y] = (R[Y])[X]. These are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials.
Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element.
Other related objects studied in abstract algebra are formal power series, which are like polynomials but may have infinite degree, and the rational functions, which are ratios of polynomials.
Further Information
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