Applications Polynomial

1 applications

1.1 calculus
1.2 abstract algebra

1.2.1 divisibility


1.3 other applications

applications
calculus

the simple structure of polynomial functions makes them quite useful in analyzing general functions using polynomial approximations. important example in calculus taylor s theorem, states every differentiable function locally looks polynomial function, , stone–weierstrass theorem, states every continuous function defined on compact interval of real axis can approximated on whole interval closely desired polynomial function.

calculating derivatives , integrals of polynomial functions particularly simple. polynomial function

∑

i
=
0


n



a

i



x

i




{\displaystyle \sum _{i=0}^{n}a_{i}x^{i}}

the derivative respect x is

∑

i
=
1


n



a

i


i

x

i
−
1




{\displaystyle \sum _{i=1}^{n}a_{i}ix^{i-1}}

and indefinite integral is

∑

i
=
0


n





a

i



i
+
1




x

i
+
1


+
c
.


{\displaystyle \sum _{i=0}^{n}{a_{i} \over i+1}x^{i+1}+c.}



abstract algebra

in abstract algebra, 1 distinguishes between polynomials , polynomial functions. polynomial f in 1 indeterminate x on ring r defined formal expression of form

f
=

a

n



x

n


+

a

n
−
1



x

n
−
1


+
⋯
+

a

1



x

1


+

a

0



x

0




{\displaystyle f=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x^{1}+a_{0}x^{0}}

where n natural number, coefficients a0, . . ., elements of r, , x formal symbol, powers x placeholders corresponding coefficients ai, given formal expression way encode sequence (a0, a1, . . .), there n such ai = 0 > n. 2 polynomials sharing same value of n considered equal if , if sequences of coefficients equal; furthermore polynomial equal polynomial greater value of n obtained adding terms in front coefficient zero. these polynomials can added adding corresponding coefficients (the rule extending terms 0 coefficients can used make sure such coefficients exist). each polynomial equal sum of terms used in formal expression, if such term aix interpreted polynomial has 0 coefficients @ powers of x other x. define multiplication, suffices distributive law describe product of 2 such terms, given rule

   elements a, b of ring r , natural numbers k , l.

thus set of polynomials coefficients in ring r forms ring, ring of polynomials on r, denoted r[x]. map r r[x] sending r rx injective homomorphism of rings, r viewed subring of r[x]. if r commutative, r[x] algebra on r.

one can think of ring r[x] arising r adding 1 new element x r, , extending in minimal way ring in x satisfies no other relations obligatory ones, plus commutation elements of r (that xr = rx). this, 1 must add powers of x , linear combinations well.

formation of polynomial ring, forming factor rings factoring out ideals, important tools constructing new rings out of known ones. instance, ring (in fact field) of complex numbers, can constructed polynomial ring r[x] on real numbers factoring out ideal of multiples of polynomial x + 1. example construction of finite fields, proceeds similarly, starting out field of integers modulo prime number coefficient ring r (see modular arithmetic).

if r commutative, 1 can associate every polynomial p in r[x], polynomial function f domain , range equal r (more 1 can take domain , range same unital associative algebra on r). 1 obtains value f(r) substitution of value r symbol x in p. 1 reason distinguish between polynomials , polynomial functions on rings different polynomials may give rise same polynomial function (see fermat s little theorem example r integers modulo p). not case when r real or complex numbers, whence 2 concepts not distinguished in analysis. more important reason distinguish between polynomials , polynomial functions many operations on polynomials (like euclidean division) require looking @ polynomial composed of expression rather evaluating @ constant value x.

divisibility

in commutative algebra, 1 major focus of study divisibility among polynomials. if r integral domain , f , g polynomials in r[x], said f divides g or f divisor of g if there exists polynomial q in r[x] such f q = g. 1 can show every 0 gives rise linear divisor, or more formally, if f polynomial in r[x] , r element of r such f(r) = 0, polynomial (x − r) divides f. converse true. quotient can computed using polynomial long division.

if f field , f , g polynomials in f[x] g ≠ 0, there exist unique polynomials q , r in f[x] with

f
=
q

g
+
r


{\displaystyle f=q\,g+r}

and such degree of r smaller degree of g (using convention polynomial 0 has negative degree). polynomials q , r uniquely determined f , g. called euclidean division, division remainder or polynomial long division , shows ring f[x] euclidean domain.

analogously, prime polynomials (more correctly, irreducible polynomials) can defined non-zero polynomials cannot factorized product of 2 non-constant polynomials. in case of coefficients in ring, non-constant must replaced non-constant or non-unit (both definitions agree in case of coefficients in field). polynomial may decomposed product of invertible constant product of irreducible polynomials. if coefficients belong field or unique factorization domain decomposition unique order of factors , multiplication of non-unit factor unit (and division of unit factor same unit). when coefficients belong integers, rational numbers or finite field, there algorithms test irreducibility , compute factorization irreducible polynomials (see factorization of polynomials). these algorithms not practicable hand-written computation, available in computer algebra system. eisenstein s criterion can used in cases determine irreducibility.

other applications

polynomials serve approximate other functions, such use of splines.

polynomials used encode information other object. characteristic polynomial of matrix or linear operator contains information operator s eigenvalues. minimal polynomial of algebraic element records simplest algebraic relation satisfied element. chromatic polynomial of graph counts number of proper colourings of graph.

the term polynomial , adjective, can used quantities or functions can written in polynomial form. example, in computational complexity theory phrase polynomial time means time takes complete algorithm bounded polynomial function of variable, such size of input.