Polynomial - Wikipedia

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of ... Polynomial FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Typeofmathematicalexpressions Forlesselementaryaspectsofthesubject,seePolynomialring. Inmathematics,apolynomialisanexpressionconsistingofindeterminates(alsocalledvariables)andcoefficients,thatinvolvesonlytheoperationsofaddition,subtraction,multiplication,andnon-negativeintegerexponentiationofvariables.Anexampleofapolynomialofasingleindeterminatexisx2−4x+7.Anexampleinthreevariablesisx3+2xyz2−yz+1. Polynomialsappearinmanyareasofmathematicsandscience.Forexample,theyareusedtoformpolynomialequations,whichencodeawiderangeofproblems,fromelementarywordproblemstocomplicatedscientificproblems;theyareusedtodefinepolynomialfunctions,whichappearinsettingsrangingfrombasicchemistryandphysicstoeconomicsandsocialscience;theyareusedincalculusandnumericalanalysistoapproximateotherfunctions.Inadvancedmathematics,polynomialsareusedtoconstructpolynomialringsandalgebraicvarieties,whicharecentralconceptsinalgebraandalgebraicgeometry. Contents 1Etymology 2Notationandterminology 3Definition 4Classification 5Arithmetic 5.1Additionandsubtraction 5.2Multiplication 5.3Composition 5.4Division 5.5Factoring 5.6Calculus 6Polynomialfunctions 6.1Graphs 7Equations 7.1Solvingequations 8Polynomialexpressions 8.1Trigonometricpolynomials 8.2Matrixpolynomials 8.3Exponentialpolynomials 9Relatedconcepts 9.1Rationalfunctions 9.2Laurentpolynomials 9.3Powerseries 10Polynomialring 10.1Divisibility 11Applications 11.1Positionalnotation 11.2Interpolationandapproximation 11.3Otherapplications 12History 12.1Historyofthenotation 13Seealso 14Notes 15References 16Externallinks Etymology[edit] Thewordpolynomialjoinstwodiverseroots:theGreekpoly,meaning"many",andtheLatinnomen,or"name".ItwasderivedfromthetermbinomialbyreplacingtheLatinrootbi-withtheGreekpoly-.Thatis,itmeansasumofmanyterms(manymonomials).Thewordpolynomialwasfirstusedinthe17thcentury.[1] Notationandterminology[edit] Thegraphofapolynomialfunctionofdegree3 Thexoccurringinapolynomialiscommonlycalledavariableoranindeterminate.Whenthepolynomialisconsideredasanexpression,xisafixedsymbolwhichdoesnothaveanyvalue(itsvalueis"indeterminate").However,whenoneconsidersthefunctiondefinedbythepolynomial,thenxrepresentstheargumentofthefunction,andisthereforecalleda"variable".Manyauthorsusethesetwowordsinterchangeably. ApolynomialPintheindeterminatexiscommonlydenotedeitherasPorasP(x).Formally,thenameofthepolynomialisP,notP(x),buttheuseofthefunctionalnotationP(x)datesfromatimewhenthedistinctionbetweenapolynomialandtheassociatedfunctionwasunclear.Moreover,thefunctionalnotationisoftenusefulforspecifying,inasinglephrase,apolynomialanditsindeterminate.Forexample,"letP(x)beapolynomial"isashorthandfor"letPbeapolynomialintheindeterminatex".Ontheotherhand,whenitisnotnecessarytoemphasizethenameoftheindeterminate,manyformulasaremuchsimplerandeasiertoreadifthename(s)oftheindeterminate(s)donotappearateachoccurrenceofthepolynomial. Theambiguityofhavingtwonotationsforasinglemathematicalobjectmaybeformallyresolvedbyconsideringthegeneralmeaningofthefunctionalnotationforpolynomials. Ifadenotesanumber,avariable,anotherpolynomial,or,moregenerally,anyexpression,thenP(a)denotes,byconvention,theresultofsubstitutingaforxinP.Thus,thepolynomialPdefinesthefunction a ↦ P ( a ) , {\displaystylea\mapstoP(a),} whichisthepolynomialfunctionassociatedtoP. Frequently,whenusingthisnotation,onesupposesthataisanumber.However,onemayuseitoveranydomainwhereadditionandmultiplicationaredefined(thatis,anyring).Inparticular,ifaisapolynomialthenP(a)isalsoapolynomial. Morespecifically,whenaistheindeterminatex,thentheimageofxbythisfunctionisthepolynomialPitself(substitutingxforxdoesnotchangeanything).Inotherwords, P ( x ) = P , {\displaystyleP(x)=P,} whichjustifiesformallytheexistenceoftwonotationsforthesamepolynomial. Definition[edit] Apolynomialexpressionisanexpressionthatcanbebuiltfromconstantsandsymbolscalledvariablesorindeterminatesbymeansofaddition,multiplicationandexponentiationtoanon-negativeintegerpower.Theconstantsaregenerallynumbers,butmaybeanyexpressionthatdonotinvolvetheindeterminates,andrepresentmathematicalobjectsthatcanbeaddedandmultiplied.Twopolynomialexpressionsareconsideredasdefiningthesamepolynomialiftheymaybetransformed,onetotheother,byapplyingtheusualpropertiesofcommutativity,associativityanddistributivityofadditionandmultiplication.Forexample ( x − 1 ) ( x − 2 ) {\displaystyle(x-1)(x-2)} and x 2 − 3 x + 2 {\displaystylex^{2}-3x+2} aretwopolynomialexpressionsthatrepresentthesamepolynomial;so,onewrites ( x − 1 ) ( x − 2 ) = x 2 − 3 x + 2. {\displaystyle(x-1)(x-2)=x^{2}-3x+2.} Apolynomialinasingleindeterminatexcanalwaysbewritten(orrewritten)intheform a n x n + a n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0 , {\displaystylea_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb+a_{2}x^{2}+a_{1}x+a_{0},} where a 0 , … , a n {\displaystylea_{0},\ldots,a_{n}} areconstantsthatarecalledthecoefficientsofthepolynomial,and x {\displaystylex} istheindeterminate.[2]Theword"indeterminate"meansthat x {\displaystylex} representsnoparticularvalue,althoughanyvaluemaybesubstitutedforit.Themappingthatassociatestheresultofthissubstitutiontothesubstitutedvalueisafunction,calledapolynomialfunction. Thiscanbeexpressedmoreconciselybyusingsummationnotation: ∑ k = 0 n a k x k {\displaystyle\sum_{k=0}^{n}a_{k}x^{k}} Thatis,apolynomialcaneitherbezeroorcanbewrittenasthesumofafinitenumberofnon-zeroterms.Eachtermconsistsoftheproductofanumber –calledthecoefficientoftheterm[a] –andafinitenumberofindeterminates,raisedtonon-negativeintegerpowers. Classification[edit] Furtherinformation:Degreeofapolynomial Theexponentonanindeterminateinatermiscalledthedegreeofthatindeterminateinthatterm;thedegreeofthetermisthesumofthedegreesoftheindeterminatesinthatterm,andthedegreeofapolynomialisthelargestdegreeofanytermwithnonzerocoefficient.[3]Becausex=x1,thedegreeofanindeterminatewithoutawrittenexponentisone. Atermwithnoindeterminatesandapolynomialwithnoindeterminatesarecalled,respectively,aconstanttermandaconstantpolynomial.[b]Thedegreeofaconstanttermandofanonzeroconstantpolynomialis0.Thedegreeofthezeropolynomial0(whichhasnotermsatall)isgenerallytreatedasnotdefined(butseebelow).[4] Forexample: − 5 x 2 y {\displaystyle-5x^{2}y} isaterm.Thecoefficientis−5,theindeterminatesarexandy,thedegreeofxistwo,whilethedegreeofyisone.Thedegreeoftheentiretermisthesumofthedegreesofeachindeterminateinit,sointhisexamplethedegreeis2+1=3. Formingasumofseveraltermsproducesapolynomial.Forexample,thefollowingisapolynomial: 3 x 2 ⏟ t e r m 1 − 5 x ⏟ t e r m 2 + 4 ⏟ t e r m 3 . {\displaystyle\underbrace{_{\,}3x^{2}}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{1}\end{smallmatrix}}\underbrace{-_{\,}5x}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{2}\end{smallmatrix}}\underbrace{+_{\,}4}_{\begin{smallmatrix}\mathrm{term}\\\mathrm{3}\end{smallmatrix}}.} Itconsistsofthreeterms:thefirstisdegreetwo,thesecondisdegreeone,andthethirdisdegreezero. Polynomialsofsmalldegreehavebeengivenspecificnames.Apolynomialofdegreezeroisaconstantpolynomial,orsimplyaconstant.Polynomialsofdegreeone,twoorthreearerespectivelylinearpolynomials,quadraticpolynomialsandcubicpolynomials.[3]Forhigherdegrees,thespecificnamesarenotcommonlyused,althoughquarticpolynomial(fordegreefour)andquinticpolynomial(fordegreefive)aresometimesused.Thenamesforthedegreesmaybeappliedtothepolynomialortoitsterms.Forexample,theterm2xinx2+2x+1isalinearterminaquadraticpolynomial. Thepolynomial0,whichmaybeconsideredtohavenotermsatall,iscalledthezeropolynomial.Unlikeotherconstantpolynomials,itsdegreeisnotzero.Rather,thedegreeofthezeropolynomialiseitherleftexplicitlyundefined,ordefinedasnegative(either−1or−∞).[5]Thezeropolynomialisalsouniqueinthatitistheonlypolynomialinoneindeterminatethathasaninfinitenumberofroots.Thegraphofthezeropolynomial,f(x)=0,isthex-axis. Inthecaseofpolynomialsinmorethanoneindeterminate,apolynomialiscalledhomogeneousofdegreenifallofitsnon-zerotermshavedegreen.Thezeropolynomialishomogeneous,and,asahomogeneouspolynomial,itsdegreeisundefined.[c]Forexample,x3y2+7x2y3−3x5ishomogeneousofdegree5.Formoredetails,seeHomogeneouspolynomial. Thecommutativelawofadditioncanbeusedtorearrangetermsintoanypreferredorder.Inpolynomialswithoneindeterminate,thetermsareusuallyorderedaccordingtodegree,eitherin"descendingpowersofx",withthetermoflargestdegreefirst,orin"ascendingpowersofx".Thepolynomial3x2-5x+4iswrittenindescendingpowersofx.Thefirsttermhascoefficient3,indeterminatex,andexponent2.Inthesecondterm,thecoefficientis−5.Thethirdtermisaconstant.Becausethedegreeofanon-zeropolynomialisthelargestdegreeofanyoneterm,thispolynomialhasdegreetwo.[6] Twotermswiththesameindeterminatesraisedtothesamepowersarecalled"similarterms"or"liketerms",andtheycanbecombined,usingthedistributivelaw,intoasingletermwhosecoefficientisthesumofthecoefficientsofthetermsthatwerecombined.Itmayhappenthatthismakesthecoefficient0.[7]Polynomialscanbeclassifiedbythenumberoftermswithnonzerocoefficients,sothataone-termpolynomialiscalledamonomial,[d]atwo-termpolynomialiscalledabinomial,andathree-termpolynomialiscalledatrinomial.Theterm"quadrinomial"isoccasionallyusedforafour-termpolynomial. Arealpolynomialisapolynomialwithrealcoefficients.Whenitisusedtodefineafunction,thedomainisnotsorestricted.However,arealpolynomialfunctionisafunctionfromtherealstotherealsthatisdefinedbyarealpolynomial.Similarly,anintegerpolynomialisapolynomialwithintegercoefficients,andacomplexpolynomialisapolynomialwithcomplexcoefficients. Apolynomialinoneindeterminateiscalledaunivariatepolynomial,apolynomialinmorethanoneindeterminateiscalledamultivariatepolynomial.Apolynomialwithtwoindeterminatesiscalledabivariatepolynomial.[2]Thesenotionsrefermoretothekindofpolynomialsoneisgenerallyworkingwiththantoindividualpolynomials;forinstance,whenworkingwithunivariatepolynomials,onedoesnotexcludeconstantpolynomials(whichmayresultfromthesubtractionofnon-constantpolynomials),althoughstrictlyspeaking,constantpolynomialsdonotcontainanyindeterminatesatall.Itispossibletofurtherclassifymultivariatepolynomialsasbivariate,trivariate,andsoon,accordingtothemaximumnumberofindeterminatesallowed.Again,sothatthesetofobjectsunderconsiderationbeclosedundersubtraction,astudyoftrivariatepolynomialsusuallyallowsbivariatepolynomials,andsoon.Itisalsocommontosaysimply"polynomialsinx,y,andz",listingtheindeterminatesallowed. Theevaluationofapolynomialconsistsofsubstitutinganumericalvaluetoeachindeterminateandcarryingouttheindicatedmultiplicationsandadditions.Forpolynomialsinoneindeterminate,theevaluationisusuallymoreefficient(lowernumberofarithmeticoperationstoperform)usingHorner'smethod: ( ( ( ( ( a n x + a n − 1 ) x + a n − 2 ) x + ⋯ + a 3 ) x + a 2 ) x + a 1 ) x + a 0 . {\displaystyle(((((a_{n}x+a_{n-1})x+a_{n-2})x+\dotsb+a_{3})x+a_{2})x+a_{1})x+a_{0}.} Arithmetic[edit] Additionandsubtraction[edit] Polynomialscanbeaddedusingtheassociativelawofaddition(groupingalltheirtermstogetherintoasinglesum),possiblyfollowedbyreordering(usingthecommutativelaw)andcombiningofliketerms.[7][8]Forexample,if P = 3 x 2 − 2 x + 5 x y − 2 {\displaystyleP=3x^{2}-2x+5xy-2} and Q = − 3 x 2 + 3 x + 4 y 2 + 8 {\displaystyleQ=-3x^{2}+3x+4y^{2}+8} thenthesum P + Q = 3 x 2 − 2 x + 5 x y − 2 − 3 x 2 + 3 x + 4 y 2 + 8 {\displaystyleP+Q=3x^{2}-2x+5xy-2-3x^{2}+3x+4y^{2}+8} canbereorderedandregroupedas P + Q = ( 3 x 2 − 3 x 2 ) + ( − 2 x + 3 x ) + 5 x y + 4 y 2 + ( 8 − 2 ) {\displaystyleP+Q=(3x^{2}-3x^{2})+(-2x+3x)+5xy+4y^{2}+(8-2)} andthensimplifiedto P + Q = x + 5 x y + 4 y 2 + 6. {\displaystyleP+Q=x+5xy+4y^{2}+6.} Whenpolynomialsareaddedtogether,theresultisanotherpolynomial.[9] Subtractionofpolynomialsissimilar. Multiplication[edit] Polynomialscanalsobemultiplied.Toexpandtheproductoftwopolynomialsintoasumofterms,thedistributivelawisrepeatedlyapplied,whichresultsineachtermofonepolynomialbeingmultipliedbyeverytermoftheother.[7]Forexample,if P = 2 x + 3 y + 5 Q = 2 x + 5 y + x y + 1 {\displaystyle{\begin{aligned}\color{Red}P&\color{Red}{=2x+3y+5}\\\color{Blue}Q&\color{Blue}{=2x+5y+xy+1}\end{aligned}}} then P Q = ( 2 x ⋅ 2 x ) + ( 2 x ⋅ 5 y ) + ( 2 x ⋅ x y ) + ( 2 x ⋅ 1 ) + ( 3 y ⋅ 2 x ) + ( 3 y ⋅ 5 y ) + ( 3 y ⋅ x y ) + ( 3 y ⋅ 1 ) + ( 5 ⋅ 2 x ) + ( 5 ⋅ 5 y ) + ( 5 ⋅ x y ) + ( 5 ⋅ 1 ) {\displaystyle{\begin{array}{rccrcrcrcr}{\color{Red}{P}}{\color{Blue}{Q}}&{=}&&({\color{Red}{2x}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{2x}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{2x}}\cdot{\color{Blue}{xy}})&+&({\color{Red}{2x}}\cdot{\color{Blue}{1}})\\&&+&({\color{Red}{3y}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{3y}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{3y}}\cdot{\color{Blue}{xy}})&+&({\color{Red}{3y}}\cdot{\color{Blue}{1}})\\&&+&({\color{Red}{5}}\cdot{\color{Blue}{2x}})&+&({\color{Red}{5}}\cdot{\color{Blue}{5y}})&+&({\color{Red}{5}}\cdot{\color{Blue}{xy}})&+&({\color{Red}{5}}\cdot{\color{Blue}{1}})\end{array}}} Carryingoutthemultiplicationineachtermproduces P Q = 4 x 2 + 10 x y + 2 x 2 y + 2 x + 6 x y + 15 y 2 + 3 x y 2 + 3 y + 10 x + 25 y + 5 x y + 5. {\displaystyle{\begin{array}{rccrcrcrcr}PQ&=&&4x^{2}&+&10xy&+&2x^{2}y&+&2x\\&&+&6xy&+&15y^{2}&+&3xy^{2}&+&3y\\&&+&10x&+&25y&+&5xy&+&5.\end{array}}} Combiningsimilartermsyields P Q = 4 x 2 + ( 10 x y + 6 x y + 5 x y ) + 2 x 2 y + ( 2 x + 10 x ) + 15 y 2 + 3 x y 2 + ( 3 y + 25 y ) + 5 {\displaystyle{\begin{array}{rcccrcrcrcr}PQ&=&&4x^{2}&+&(10xy+6xy+5xy)&+&2x^{2}y&+&(2x+10x)\\&&+&15y^{2}&+&3xy^{2}&+&(3y+25y)&+&5\end{array}}} whichcanbesimplifiedto P Q = 4 x 2 + 21 x y + 2 x 2 y + 12 x + 15 y 2 + 3 x y 2 + 28 y + 5. {\displaystylePQ=4x^{2}+21xy+2x^{2}y+12x+15y^{2}+3xy^{2}+28y+5.} Asintheexample,theproductofpolynomialsisalwaysapolynomial.[9][4] Composition[edit] Givenapolynomial f {\displaystylef} ofasinglevariableandanotherpolynomialgofanynumberofvariables,thecomposition f ∘ g {\displaystylef\circg} isobtainedbysubstitutingeachcopyofthevariableofthefirstpolynomialbythesecondpolynomial.[4]Forexample,if f ( x ) = x 2 + 2 x {\displaystylef(x)=x^{2}+2x} and g ( x ) = 3 x + 2 {\displaystyleg(x)=3x+2} then ( f ∘ g ) ( x ) = f ( g ( x ) ) = ( 3 x + 2 ) 2 + 2 ( 3 x + 2 ) . {\displaystyle(f\circg)(x)=f(g(x))=(3x+2)^{2}+2(3x+2).} Acompositionmaybeexpandedtoasumoftermsusingtherulesformultiplicationanddivisionofpolynomials.Thecompositionoftwopolynomialsisanotherpolynomial.[10] Division[edit] Thedivisionofonepolynomialbyanotherisnottypicallyapolynomial.Instead,suchratiosareamoregeneralfamilyofobjects,calledrationalfractions,rationalexpressions,orrationalfunctions,dependingoncontext.[11]Thisisanalogoustothefactthattheratiooftwointegersisarationalnumber,notnecessarilyaninteger.[12][13]Forexample,thefraction1/(x2+1)isnotapolynomial,anditcannotbewrittenasafinitesumofpowersofthevariablex. Forpolynomialsinonevariable,thereisanotionofEuclideandivisionofpolynomials,generalizingtheEuclideandivisionofintegers.[e]Thisnotionofthedivisiona(x)/b(x)resultsintwopolynomials,aquotientq(x)andaremainderr(x),suchthata=bq+randdegree(r)