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Polynomial functions mcTYpolynomial Manycommonfunctionsarepolynomialfunctions

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Inthisuni twedescribepolynomialfunctions andlookatsomeoftheirproperties Inordertomasterthetechniquesexplainedhereitisvitalt hatyouundertakeplentyofpractice exercisessothattheybecomesecondnature Afterreadingthistextandorviewingthevideotutorialo nthist

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Polynomial functions mcTYpolynomial Manycommonfunctionsarepolynomialfunctions






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Polynomialfunctions mc-TY-polynomial-2009-1 Manycommonfunctionsarepolynomialfunctions.Inthisunitwedescribepolynomialfunctionsandlookatsomeoftheirproperties.Inordertomasterthetechniquesexplainedhereitisvitalthatyouundertakeplentyofpracticeexercisessothattheybecomesecondnature.Afterreadingthistext,and/orviewingthevideotutorialonthistopic,youshouldbeableto:recognisewhenaruledescribesapolynomialfunction,andwritedownthedegreeofthepolynomial,recognizethetypicalshapesofthegraphsofpolynomials,ofdegreeupto4,understandwhatismeantbythemultiplicityofarootofapolynomial,sketchthegraphofapolynomial,givenitsexpressionasaproductoflinearfactors.Contents1.Introduction22.Whatisapolynomial?23.Graphsofpolynomialfunctions34.Turningpointsofpolynomialfunctions65.Rootsofpolynomialfunctions7 www.mathcentre.ac.uk1c\rmathcentre2009 1.IntroductionApolynomialfunctionisafunctionsuchasaquadratic,acubic,aquartic,andsoon,involvingonlynon-negativeintegerpowersofx.Wecangiveageneralde ntionofapolynomial,andde neitsdegree.2.Whatisapolynomial?Apolynomialofdegreenisafunctionoftheformf(x)=anxn+an1xn1+:::+a2x2+a1x+a0wherethea'sarerealnumbers(sometimescalledthecoecientsofthepolynomial).Althoughthisgeneralformulamightlookquitecomplicated,particularexamplesaremuchsimpler.Forexample,f(x)=4x33x2+2isapolynomialofdegree3,as3isthehighestpowerofxintheformula.Thisiscalledacubicpolynomial,orjustacubic.Andf(x)=x74x5+1isapolynomialofdegree7,as7isthehighestpowerofx.Noticeherethatwedon'tneedeverypowerofxupto7:weneedtoknowonlythehighestpowerofxto ndoutthedegree.Anexampleofakindyoumaybefamiliarwithisf(x)=4x22x4whichisapolynomialofdegree2,as2isthehighestpowerofx.Thisiscalledaquadratic.Functionscontainingotheroperations,suchassquareroots,arenotpolynomials.Forexample,f(x)=4x3+p x1isnotapolynomialasitcontainsasquareroot.Andf(x)=5x42x2+3=xisnotapolynomialasitcontainsa`dividebyx'. KeyPointApolynomialisafunctionoftheformf(x)=anxn+an1xn1+:::+a2x2+a1x+a0:Thedegreeofapolynomialisthehighestpowerofxinitsexpression.Constant(non-zero)polynomials,linearpolynomials,quadratics,cubicsandquarticsarepolynomialsofdegree0,1,2,3and4respectively.Thefunctionf(x)=0isalsoapolynomial,butwesaythatitsdegreeis`unde ned'. www.mathcentre.ac.uk2c\rmathcentre2009 3.GraphsofpolynomialfunctionsWehavemetsomeofthebasicpolynomialsalready.Forexample,f(x)=2isaconstantfunctionandf(x)=2x+1isalinearfunction. f(x)x 1 2 ) = 2) = 2 + 1 Itisimportanttonoticethatthegraphsofconstantfunctionsandlinearfunctionsarealwaysstraightlines.Wehavealreadysaidthataquadraticfunctionisapolynomialofdegree2.Herearesomeexamplesofquadraticfunctions:f(x)=x2;f(x)=2x2;f(x)=5x2:Whatistheimpactofchangingthecoecientofx2aswehavedoneintheseexamples?Onewayto ndoutistosketchthegraphsofthefunctions. f(x)x f(x) = x2 ) = 2 f(x) = 5x2 Youcanseefromthegraphthat,asthecoecientofx2isincreased,thegraphisstretchedvertically(thatis,intheydirection).Whatwillhappenifthecoecientisnegative?Thiswillmeanthatallofthepositivef(x)valueswillnowbecomenegative.Sowhatwillthegraphsofthefunctionslooklike?Thefunctionsarenowf(x)=x2;f(x)=2x2;f(x)=5x2: www.mathcentre.ac.uk3c\rmathcentre2009 f(x)x f(x) = -x2 f(x) = -2x2 f(x) = -5x2 Noticeherethatallofthesegraphshaveactuallybeenre\rectedinthex-axis.Thiswillalwayshappenforfunctionsofanydegreeiftheyaremultipliedby1.Nowletuslookatsomeotherquadraticfunctionstoseewhathappenswhenwevarythecoecientofx,ratherthanthecoecientofx2.Weshalluseatableofvaluesinordertoplotthegraphs,butweshall llinonlythosevaluesneartheturningpointsofthefunctions. x 5 4 3 2 1 0 1 2 x2+x 6 2 0 0 2 6 x2+4x 0 3 4 3 0 x2+6x 5 8 9 8 5 Youcanseethesymmetryineachrowofthetable,demonstratingthatwehaveconcentratedontheregionaroundtheturningpointofeachfunction.Wecannowusethesevaluestoplotthegraphs. f(x)x + 4 f(x) = x2 + 6x + Asyoucansee,increasingthepositivecoecientofxinthispolynomialmovesthegraphdownandtotheleft. www.mathcentre.ac.uk4c\rmathcentre2009 Whathappensifthecoecientofxisnegative? x 2 1 0 1 2 3 4 5 x2x 6 2 0 0 2 6 x24x 0 3 4 3 0 x26x 5 8 9 8 5 Againwecanusethesetablesofvaluestoplotthegraphsofthefunctions. f(x)x f(x) = x2 - 4x f(x) = x2 - 6x f(x) = x2 - x Asyoucansee,increasingthenegativecoecientofx(inabsoluteterms)movesthegraphdownandtotheright.Sonowweknowwhathappenswhenwevarythex2coecient,andwhathappenswhenwevarythexcoecient.Butwhathappenswhenwevarytheconstanttermattheendofourpolynomial?Wealreadyknowwhatthegraphofthefunctionf(x)=x2+xlookslike,sohowdoesthisdi erfromthegraphofthefunctionsf(x)=x2+x+1,orf(x)=x2+x+5,orf(x)=x2+x4?Asusual,atableofvaluesisagoodplacetostart. x 2 1 0 1 2 x2+x 2 0 0 2 6 x2+x+1 3 1 1 3 7 x2+x+5 7 5 5 7 11 x2+x4 2 4 4 2 2 Ourtableofvaluesisparticularlyeasytocompletesincewecanuseouranswersfromthex2+xcolumnto ndeverythingelse.Wecanusethesetablesofvaluestoplotthegraphsofthefunctions. www.mathcentre.ac.uk5c\rmathcentre2009 f(x)x + + x + + Aswecanseestraightaway,varyingtheconstanttermtranslatesthex2+xcurvevertically.Furthermore,thevalueoftheconstantisthepointatwhichthegraphcrossesthef(x)axis.4.TurningpointsofpolynomialfunctionsAturningpointofafunctionisapointwherethegraphofthefunctionchangesfromslopingdownwardstoslopingupwards,orviceversa.Sothegradientchangesfromnegativetopositive,orfrompositivetonegative.Generallyspeaking,curvesofdegreencanhaveupto(n1)turningpoints.Forinstance,aquadratichasonlyoneturningpoint. Acubiccouldhaveuptotwoturningpoints,andsowouldlooksomethinglikethis. However,somecubicshavefewerturningpoints:forexamplef(x)=x3.Butnocubichasmorethantwoturningpoints. www.mathcentre.ac.uk6c\rmathcentre2009 Inthesameway,aquarticcouldhaveuptothreeturningturningpoints,andsowouldlooksome-thinglikethis. Again,somequarticshavefewerturningpoints,butnonehasmore. KeyPointApolynomialofdegreencanhaveupto(n1)turningpoints. 5.RootsofpolynomialfunctionsYoumayrecallthatwhen(xa)(xb)=0,weknowthataandbarerootsofthefunctionf(x)=(xa)(xb).Nowwecanusetheconverseofthis,andsaythatifaandbareroots,thenthepolynomialfunctionwiththeserootsmustbef(x)=(xa)(xb),oramultipleofthis.Forexample,ifaquadratichasrootsx=3andx=2,thenthefunctionmustbef(x)=(x3)(x+2),oraconstantmultipleofthis.Thiscanbeextendedtopolynomialsofanydegree.Forexample,iftherootsofapolynomialarex=1,x=2,x=3,x=4,thenthefunctionmustbef(x)=(x1)(x2)(x3)(x4);oraconstantmultipleofthis.Letusalsothinkaboutthefunctionf(x)=(x2)2.Wecanseestraightawaythatx2=0,sothatx=2.Forthisfunctionwehaveonlyoneroot.Thisiswhatwecallarepeatedroot,andarootcanberepeatedanynumberoftimes.Forexample,f(x)=(x2)3(x+4)4hasarepeatedrootx=2,andanotherrepeatedrootx=4.Wesaythattherootx=2hasmultiplicity3,andthattherootx=4hasmultiplicity4.Theusefulthingaboutknowingthemultiplicityofarootisthatithelpsuswithsketchingthegraphofthefunction.Ifthemultiplicityofarootisoddthenthegraphcutsthroughthex-axisatthepoint(x;0).Butifthemultiplicityiseventhenthegraphjusttouchesthex-axisatthepoint(x;0).Forexample,takethefunctionf(x)=(x3)2(x+1)5(x2)3(x+2)4:Therootx=3hasmultiplicity2,sothegraphtouchesthex-axisat(3;0). www.mathcentre.ac.uk7c\rmathcentre2009 Therootx=1hasmultiplicity5,sothegraphcrossesthex-axisat(1;0).Therootx=2hasmultiplicity3,sothegraphcrossesthex-axisat(2;0).Therootx=2hasmultiplicity4,sothegraphtouchesthex-axisat(2;0).Totakeanotherexample,supposewehavethefunctionf(x)=(x2)2(x+1).Wecanseethatthelargestpowerofxis3,andsothefunctionisacubic.Weknowthepossiblegeneralshapesofacubic,andasthecoecientofx3ispositivethecurvemustgenerallyincreasetotherightanddecreasetotheleft.Wecanalsoseethattherootsofthefunctionarex=2andx=1.Therootx=2hasevenmultiplicityandsothecurvejusttouchesthex-axishere,whilstx=1hasoddmultiplicityandsoherethecurvecrossesthex-axis.Thismeanswecansketchthegraphasfollows. 2-1x f(x) KeyPointThenumberaisarootofthepolynomialfunctionf(x)iff(a)=0,andthisoccurswhen(xa)isafactoroff(x).Ifaisarootoff(x),andif(xa)misafactoroff(x)but(xa)m+1isnotafactor,thenwesaythattheroothasmultiplicitym.Atarootofoddmultiplicitythegraphofthefunctioncrossesthex-axis,whereasatarootofevenmultiplicitythegraphtouchesthex-axis. Exercises1.Whatisapolynomialfunction?2.Whichofthefollowingfunctionsarepolynomialfunctions?(a)f(x)=4x2+2(b)f(x)=3x32x+p x(c)f(x)=124x5+3x2(d)f(x)=sinx+1(e)f(x)=3x122=x(f)f(x)=3x112x12 www.mathcentre.ac.uk8c\rmathcentre2009 3.Writedownoneexampleofeachofthefollowingtypesofpolynomialfunction:(a)cubic(b)linear(c)quartic(d)quadratic4.Sketchthegraphsofthefollowingfunctionsonthesameaxes:(a)f(x)=x2(b)f(x)=4x2(c)f(x)=x2(d)f(x)=4x25.Considerafunctionoftheformf(x)=x2+ax,wherearepresentsarealnumber.Thegraphofthisfunctionisrepresentedbyaparabola.(a)Whena�0,whathappenstotheparabolaasaincreases?(b)Whena0,whathappenstotheparabolaasadecreases?6.Writedownthemaximumnumberofturningpointsonthegraphofapolynomialfunctionofdegree:(a)2(b)3(c)12(d)n7.Writedownapolynomialfunctionwithroots:(a)1;2;3;4(b)2;4(c)12;1;68.Writedowntherootsandidentifytheirmultiplicityforeachofthefollowingfunctions:(a)f(x)=(x2)3(x+4)4(b)f(x)=(x1)(x+2)2(x4)39.Sketchthefollowingfunctions:(a)f(x)=(x2)2(x+1)(b)f(x)=(x1)2(x+3)Answers1.Apolynomialfunctionisafunctionthatcanbewrittenintheformf(x)=anxn+an1xn1+an2xn2+:::+a2x2+a1x+a0;whereeacha0,a1,etc.representsarealnumber,andwherenisanaturalnumber(including0).2.(a)f(x)=4x2+2isapolynomial(b)f(x)=3x32x+p xisnotapolynomial,becauseofp x(c)f(x)=124x5+3x2isapolynomial(d)f(x)=sinx+1isnotapolynomial,becauseofsinx(e)f(x)=3x122=xisnotapolynomial,becauseof2=x(f)f(x)=3x112x12isapolynomial3.(a)Thehighestpowerofxmustbe3,soexamplesmightbef(x)=x32x+1orf(x)=x32.(b)Thehighestpowerofxmustbe1,soexamplesmightbef(x)=xorf(x)=6x5.(c)Thehighestpowerofxmustbe4,soexamplesmightbef(x)=x43x3+2xorf(x)=x4x5.(d)Thehighestpowerofxmustbe2,soexamplesmightbef(x)=x2orf(x)=x25.4. www.mathcentre.ac.uk9c\rmathcentre2009 f(x)x f(x) = 4x2f(x) = -4x2 f(x) = x2 f(x) = -x2 5.(a)Whena�0,theparabolamovesdownandtotheleftasaincreases.(b)Whena0,theparabolamovesdownandtotherightasadecreases.6.(a)1turningpoint(b)2turningpoints(c)11turningpoints(d)(n1)turningpoints.7.(a)f(x)=(x1)(x2)(x3)(x4)oramultiple(b)f(x)=(x2)(x+4)oramultiple(c)f(x)=(x12)(x+1)(x+6)oramultiple.8.(a)x=2oddmultiplicityx=4evenmultiplicity(b)x=1oddmultiplicityx=2evenmultiplicityx=4oddmultiplicity www.mathcentre.ac.uk10c\rmathcentre2009 9) f(x)x ) = ( ) = ( www.mathcentre.ac.uk11c\rmathcentre2009