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Polynomial

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INTRODUCTION In mathematics, a polynomial (from Greek

poly, "many" and medieval atin !inomim,"!inomial"#$% #&% #'% is an e)pression of *nitelen+th constrcted from varia!les (also knon

as indeterminates and constants, sin+ onlythe operationstheory-

of addition, s!traction, mltiplication, andnon.ne+ative inte+er e)ponents- The term

"polynomial" can also !e sed as an ad/ective,for 0antities that can !e e)pressed as apolynomial of some parameter, as in"polynomial time" hich is sed incomptational comple)ity

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O12R1I23

polynomial is either 4ero, or can !e ritten asthe sm of one or more non.4ero terms- Thenm!er of terms is *nite- These terms consistof a constant (called the coe5cient of the

term hich may !e mltiplied !y a *nitenm!er of varia!les (sally represented !yletters- 2ach varia!le may have an e)ponentthat is a non.ne+ative inte+er, i-e-, a natral

nm!er- The e)ponent on a varia!le in a termis called the de+ree of that varia!le in thatterm, the de+ree of the term is the sm of thede+rees of the varia!les in that term, and thede+ree of a polynomial is the lar+est de+reeof any one term- 6ince ) 7 )$, the de+ree of

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:olynomial fnctions 8 polynomial fnction is a fnction that can !e de*ned !y evalatin+ a

polynomial- 8 fnction ; of one ar+ment is called a polynomial fnction if itsatis*es

for all ar+ments ), here n is a non.ne+ative inte+er and a9, a$,a&, ---, an

are constant coe5cients-

$ (see Che!yshev polynomials-

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:olynomial e0ations 8 polynomial e0ation is an e0ation in hich apolynomial is set e0al to another polynomial-

is a polynomial e0ation- In case of a polynomial e0ationthe varia!le is considered an nknon, and one seeks to*nd the possi!le vales for hich !oth mem!ers of thee0ation evalate to the same vale (in +eneral morethan one soltion may e)ist- 8 polynomial e0ation is to!e contrasted ith a polynomial identity like () ? y() @ y

7 )& @ y&, here !oth mem!ers represent the samepolynomial in diAerent forms, and as a conse0ence anyevalation of !oth mem!ers ill +ive a valid e0ality- Thismeans that a polynomial identity is a polynomial e0ationfor hich all possi!le vales of the nknons are soltions

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2lementary properties ofpolynomials 8 sm of polynomials is a polynomial-

8 prodct of polynomials is a polynomial-

8 composition of to polynomials is a polynomial,hich is o!tained !y s!stittin+ a varia!le of the

*rst polynomial !y the second polynomial- The derivative of the polynomial an)n ? an.$)n.$ ?--- ? a&)& ? a$) ? a9 is the polynomial nan)n.$ ? (n.$an.$)n.& ? --- ? &a&) ? a$- If the set of thecoe5cients does not contain the inte+ers (for

e)ample if the coe5cients are inte+ers modlo someprime nm!er p, then kak shold !e interpreted asthe sm of ak ith itself, k times-

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2valation of apolynomial 2valation of a polynomial consists of assi+nin+ a

nm!er to each varia!le and carryin+ ot the indicatedmltiplications and additions- 8ctal evalation issally more e5cient sin+ the Borner scheme

In elementary al+e!ra, methods are +iven for solvin+all *rst de+ree and second de+ree polynomiale0ations in one varia!le- In the case of polynomiale0ations, the varia!le is often called an nknon- The

nm!er of soltions may not e)ceed the de+ree, andill e0al the de+ree hen mltiplicity of soltions andcomple) nm!er soltions are conted- This fact iscalled the fndamental theorem of al+e!ra-

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BI6TOR Determinin+ the roots of polynomials, or "solvin+

al+e!raic e0ations", is amon+ the oldestpro!lems in mathematics- Boever, the ele+antand practical notation e se today only

developed !e+innin+ in the $Eth centry- Feforethat, e0ations ere ritten ot in ords-

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Notation

The earliest knon se of the e0al si+n is inRo!ert Recorde=s The 3hetstone of 3itte, $EEH-The si+ns ? for addition, > for s!traction, and these of a letter for an nknon appear in ichael

6tifel=s 8rithemetica inte+ra, $EJJ- RenK Descartes,in a +Kometrie, $L'H, introdced the concept ofthe +raph of a polynomial e0ation- Be poplari4edthe se of letters from the !e+innin+ of thealpha!et to denote constants and letters from the

end of the alpha!et to denote varia!les, as can !eseen a!ove, in the +eneral formla for a polynomialin one varia!le, here the a =s denote constantsand ) denotes a varia!le- Descartes introdced these of sperscripts to denote e)ponents as ell-#L%

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Nm!er of varia!les

One classi*cation of polynomials is !ased on the nm!er ofdistinct varia!les- 8 polynomial in one varia!le is called anivariate polynomial, a polynomial in more than one varia!le iscalled a mltivariate polynomial- These notions refer more to thekind of polynomials one is +enerally orkin+ ith than to

individal polynomialsM for instance hen orkin+ ith nivariatepolynomials one does not e)clde constant polynomials (hichmay reslt, for instance, from the s!traction of non.constantpolynomials, altho+h strictly speakin+ constant polynomials donot contain any varia!les at all- It is possi!le to frther classifymltivariate polynomials as !ivariate, trivariate, and so on,

accordin+ to the ma)imm nm!er of varia!les alloed- 8+ain, sothat the set of o!/ects nder consideration !e closed nders!traction, a stdy of trivariate polynomials sally allos!ivariate polynomials, and so on- It is common, also, to say simply"polynomials in ), y, and 4", listin+ the varia!les alloed- In thiscase, )y is alloed

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2)tensions of the concept of a polynomial

:olynomials can involve more than one varia!le, in hich they arecalled mltivariate- Rin+s of polynomials in a *nite nm!er ofvaria!les are of fndamental importance in al+e!raic +eometryhich stdies the simltaneos 4ero sets of several schmltivariate polynomials- These rin+s can alternatively !e

constrcted !y repeatin+ the constrction of nivariatepolynomials ith as coe5cient rin+ another rin+ of polynomialsths the rin+ R#,% of polynomials in and can !e vieed asthe rin+ (R#%#% of polynomials in ith as coe5cientspolynomials in , or as the rin+ (R#%#% of polynomials in ithas coe5cients polynomials in - These identi*cations are

compati!le ith arithmetic operations (they are isomorphisms ofrin+s, !t some notions sch as de+ree or hether a polynomialis considered monic can chan+e !eteen these points of vie-One can constrct rin+s of polynomials in in*nitely manyvaria!les, !t since polynomials are (*nite e)pressions, anyindividal polynomial can only contain *nitely many varia!les-

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DI1I6ION 8GORITB

If f() and are polynomials, and the de+ree of d()is less than or e0al to the de+ree of f(), thenthere e)ist ni0e polynomials 0() and r(), sothat

and so that the de+ree of r() is less than the

de+ree of d()- In the special case here r()79,e say that d() divides evenly into f()-