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In mathematics, an integer-valued polynomial ( also known as a numerical polynomial ) P ( t ) is a polynomial whose value P ( n ) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true.

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## Some Related Sentences

mathematics and polynomial

**In**

__mathematics__

**,**

**an**algebraic number

**is**

**a**number that

**is**

**a**root of

**a**non-zero

__polynomial__in one variable

**with**rational

**coefficients**

**(**or equivalently — by clearing denominators —

**with**

**integer**

**coefficients**).

Initially

**a**study of systems of__polynomial__equations in several variables**,****the**subject of algebraic geometry starts where equation solving leaves off**,**and it becomes even more important to understand**the**intrinsic properties of**the**totality of solutions of**a**system of equations**,**than to find**a**specific solution ; this leads into some of**the**deepest areas in all of__mathematics__**,**both conceptually and in terms of technique**.****In**

__mathematics__

**,**

**a**Diophantine equation

**is**

**an**indeterminate

__polynomial__equation that allows

**the**variables to be integers only

**.**

**In**

__mathematics__

**,**Horner's method

**(**

**also**

**known**

**as**Horner scheme in

**the**UK or Horner's rule in

**the**U

**.**S .)

**is**either of two things:

**(**i

**)**

**an**algorithm

**for**calculating polynomials

**,**which consists in transforming

**the**monomial form into

**a**computationally efficient form ; or

**(**ii

**)**

**a**method

**for**approximating

**the**roots of

**a**

__polynomial__

**.**

**In**

__mathematics__

**,**

**a**

__polynomial__

**is**

**an**expression of finite length constructed from variables

**(**

**also**called indeterminates

**)**and constants

**,**using only

**the**operations of addition

**,**subtraction

**,**multiplication

**,**and non-negative

**integer**exponents

**.**

**In**advanced

__mathematics__

**,**polynomials are used to construct

__polynomial__rings

**,**

**a**central concept in abstract algebra and algebraic geometry

**.**

**In**

__mathematics__

**,**

**a**transcendental number

**is**

**a**

**(**possibly complex

**)**number that

**is**

**not**algebraic — that

**is**

**,**it

**is**

**not**

**a**root of

**a**non-constant

__polynomial__equation

**with**rational

**coefficients**

**.**

**In**

__mathematics__

**,**

**a**Hurwitz

__polynomial__

**,**named after Adolf Hurwitz

**,**

**is**

**a**

__polynomial__

**whose**

**coefficients**are positive real numbers and

**whose**zeros are located in

**the**left half-plane of

**the**complex plane

**,**that

**is**

**,**

**the**real part of

**every**zero

**is**negative

**.**

" One such example of his impact on pure

__mathematics__**is**his framework**for**understanding**the**Jones__polynomial__using Chern – Simons theory**.****In**

__mathematics__

**,**factorization

**(**

**also**factorisation in British English

**)**or factoring

**is**

**the**decomposition of

**an**object

**(**

**for**example

**,**

**a**number

**,**

**a**

__polynomial__

**,**or

**a**matrix

**)**into

**a**product of other objects

**,**or factors

**,**which when multiplied together give

**the**original

**.**

**In**

__mathematics__

**,**

**a**Diophantine equation

**is**

**an**equation of

**the**form

**P**

**(**x < sub > 1 </ sub >, ..., x < sub > j </ sub >, y < sub > 1 </ sub >, ..., y < sub > k </ sub >)= 0

**(**usually abbreviated

**P**(,)= 0

**)**where

**P**(,)

**is**

**a**

__polynomial__

**with**

**integer**

**coefficients**

**.**

**In**

__mathematics__

**,**

**a**quadric

**,**or quadric surface

**,**

**is**any D-dimensional hypersurface in

**(**D + 1 )- dimensional space defined

**as**

**the**locus of zeros of

**a**quadratic

__polynomial__

**.**

**In**

__mathematics__

**,**if L

**is**

**a**field extension of K

**,**then

**an**element

**a**of L

**is**called

**an**algebraic element over K

**,**or just algebraic over K

**,**if there exists some non-zero

__polynomial__g

**(**x

**)**

**with**

**coefficients**in K such that g

**(**

**a**)= 0

**.**

**In**

__mathematics__

**,**

**a**

__polynomial__

**is**said to be irreducible if it cannot be factored into

**the**product of two or more non-trivial polynomials

**whose**

**coefficients**are of

**a**specified type

**.**

**In**

__mathematics__

**,**

**the**Hermite polynomials are

**a**classical orthogonal

__polynomial__sequence that arise in probability

**,**such

**as**

**the**Edgeworth series ; in combinatorics

**,**

**as**

**an**example of

**an**Appell sequence

**,**obeying

**the**umbral calculus ; in

**numerical**analysis

**as**Gaussian quadrature ; in finite element methods

**as**Shape Functions

**for**beams ; and in physics

**,**where they give rise to

**the**eigenstates of

**the**quantum harmonic oscillator

**.**

**In**

__mathematics__

**,**

**a**

__polynomial__sequence

**is**

**a**sequence of polynomials indexed by

**the**nonnegative integers 0

**,**1

**,**2

**,**3

**,**..., in which each index

**is**equal to

**the**degree of

**the**corresponding

__polynomial__

**.**

**In**

__mathematics__before

**the**1970s

**,**

**the**term umbral calculus referred to

**the**surprising similarity between seemingly unrelated

__polynomial__equations and certain shadowy techniques used to ' prove ' them

**.**

mathematics and also

Blind students

__also__complete mathematical assignments using**a**braille-writer and Nemeth code**(****a**type of braille code**for**__mathematics__**)****but**large multiplication and long division problems can be long and difficult**.**
The axiom of choice has

__also__been thoroughly studied in**the**context of constructive__mathematics__**,**where non-classical logic**is**employed**.**
Although

**the**axiom of countable choice in particular**is**commonly used in constructive__mathematics__**,**its use has__also__been questioned**.**
It

**is**__also__commonly used in__mathematics__in algebraic solutions representing quantities such**as**angles**.**
The term may be

__also__used loosely or metaphorically to denote highly skilled people in any non -" art " activities**,****as**well — law**,**medicine**,**mechanics**,**or__mathematics__**,****for**example**.**
Despite being quite religious

**,**he was__also__interested in__mathematics__and science**,**and sometimes**is**claimed to have contradicted**the**teachings of**the**Church in favour of scientific theories**.**
He was educated at

**the**Collège des Quatre-Nations**(**__also__**known****as**Collège Mazarin**)**from 1754 to 1761**,**studying chemistry**,**botany**,**astronomy**,**and__mathematics__**.**
His criticisms of

**the**scientific community**,**and especially of several__mathematics__circles**,**are__also__contained in**a**letter**,**written in 1988**,**in which he states**the**reasons**for**his refusal of**the**Crafoord Prize**.**
He

**is**__also__noted**for**his mastery of abstract approaches to__mathematics__and his perfectionism in matters of formulation and presentation**.****In**

__mathematics__

**,**

**the**axiom of regularity

**(**

__also__

**known**

**as**

**the**axiom of foundation

**)**

**is**one of

**the**axioms of Zermelo – Fraenkel set theory and was introduced by

**.**

**In**classical

__mathematics__

**,**analytic geometry

**,**

__also__

**known**

**as**coordinate geometry

**,**or Cartesian geometry

**,**

**is**

**the**study of geometry using

**a**coordinate system and

**the**principles of algebra and analysis

**.**

He won

**a**scholarship to**the**University and majored in__mathematics__**,**and__also__studied astronomy**,**physics and chemistry**.**
His father

**,**Étienne Pascal**(**1588 – 1651 ), who__also__had**an**interest in science and__mathematics__**,**was**a**local judge and member of**the**" Noblesse de Robe ".**In**chemistry

**,**physics

**,**and

__mathematics__

**,**

**the**Boltzmann distribution

**(**

__also__called

**the**Gibbs Distribution

**)**

**is**

**a**certain distribution function or probability measure

**for**

**the**distribution of

**the**states of

**a**system

**.**

Bioinformatics

__also__deals**with**algorithms**,**databases and information systems**,**web technologies**,**artificial intelligence and soft computing**,**information and computation theory**,**structural biology**,**software engineering**,**data mining**,**image processing**,**modeling and simulation**,**discrete__mathematics__**,**control and system theory**,**circuit theory**,**and statistics**.****In**

__mathematics__

**,**especially functional analysis

**,**

**a**Banach algebra

**,**named after Stefan Banach

**,**

**is**

**an**associative algebra A over

**the**real or complex numbers which at

**the**same time

**is**

__also__

**a**Banach space

**.**

The term can

__also__be applied to some degree to functions in__mathematics__**,**referring to**the**anatomy of curves**.**
Combinatorial problems arise in many areas of pure

__mathematics__**,**notably in algebra**,**probability theory**,**topology**,**and geometry**,**and combinatorics__also__has many applications in optimization**,**computer science**,**ergodic theory and statistical physics**.**
It has

__also__given rise to**a**new theory of**the**philosophy of__mathematics__**,**and many theories of artificial intelligence**,**persuasion and coercion**.**
Most undergraduate programs emphasize

__mathematics__and physics**as**well**as**chemistry**,**partly because chemistry**is**__also__**known****as**"**the**central science ", thus chemists ought to have**a**well-rounded knowledge about science**.**
Category theory

**is****an**area of study in__mathematics__that examines in**an**abstract way**the**properties of particular mathematical concepts**,**by formalising them**as**collections of objects and arrows**(**__also__called morphisms**,**although this term__also__has**a**specific**,**non category-theoretical meaning ), where these collections satisfy some basic conditions**.**0.147 seconds.