Polynomial

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders.

Abstract algebra

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: Main article: polynomial ring.

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In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form

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: f = a_n X^n + a_{n - 1} X^{n - 1} + cdots + a_1 X + a_0

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where the coefficients a0, ..., an are elements of some ring R and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal. Polynomials with coefficients in R can be added by simply adding corresponding coefficients and multiplied using the distributive law and the rules

Related Topics:
Ring - Distributive law

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:

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X ; a = a ; X   for all elements a of the ring R

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:

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X^k ; X^l = X^{k+l}   for all natural numbers k and l.

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One can then check that the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R. If R is commutative, then R is an algebra over R.

Related Topics:
Commutative - Algebra

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One can think of the ring R as arising from R by adding one new element X to R and only requiring that X commute with all elements of R. In order for R to form a ring, all sums of powers of X have to be included as well. Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the clean construction of finite fields involves the use of those operations, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).

Related Topics:
Ideals - Finite field - Prime number - Modular arithmetic

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To every polynomial f in R, one can associate a polynomial function with domain and range equal to R. One obtains the value of this function for a given argument r by everywhere replacing the symbol X in f's expression by r. The reason that algebraists have to distinguish between polynomials and polynomial functions is that over some rings R (for instance, over finite fields), two different polynomials may give rise to the same polynomial function. This is not the case over the real or complex numbers and therefore many analysts often don't separate the two concepts.

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