Polynomial functions mc-ty-polynomial-2009-1 Many common functions are polynomial functions. In this unit we describe polynomial functions and look at some of their properties. In order to master the techniques eplained here it is vital that you undertake plenty of practice eercises so that they become second nature. Afterreadingthistet,and/orviewingthevideotutorialonthistopic,youshouldbeableto: recognisewhenaruledescribesapolynomialfunction,andwritedownthedegreeofthe polynomial, recognizethetypicalshapesofthegraphsofpolynomials,ofdegreeupto4, understandwhatismeantbythemultiplicityofarootofapolynomial, sketchthegraphofapolynomial,givenitsepressionasaproductoflinearfactors. Contents 1. Introduction 2 2. What is a polynomial? 2 3. Graphs of polynomial functions 3 4. Turning points of polynomial functions 6 5. Roots of polynomial functions 7 www.mathcentre.ac.uk 1 c mathcentre 2009
1. Introduction Apolynomialfunctionisafunctionsuchasaquadratic,acubic,aquartic,andsoon,involving onlynon-negativeintegerpowersof. Wecangiveageneraldefintionofapolynomial,and define its degree. 2. What is a polynomial? Apolynomialofdegree nisafunctionoftheform f() = a n n + a n 1 n 1 +... + a 2 2 + a 1 + a 0 where the a s are real numbers(sometimes called the coefficients of the polynomial). Although this general formula might look quite complicated, particular eamples are much simpler. For eample, f() = 4 3 3 2 + 2 isapolynomialofdegree3,as3isthehighestpowerof intheformula.thisiscalledacubic polynomial, or just a cubic. And f() = 7 4 5 + 1 isapolynomialofdegree7,as7isthehighestpowerof.noticeherethatwedon tneedevery powerof upto7: weneedtoknowonlythehighestpowerof tofindoutthedegree. An eampleofakindyoumaybefamiliarwithis f() = 4 2 2 4 whichisapolynomialofdegree2,as2isthehighestpowerof.thisiscalledaquadratic. Functions containing other operations, such as square roots, are not polynomials. For eample, f() = 4 3 + 1 isnotapolynomialasitcontainsasquareroot.and f() = 5 4 2 2 + 3/ isnotapolynomialasitcontainsa divideby. Apolynomialisafunctionoftheform Key Point f() = a n n + a n 1 n 1 +... + a 2 2 + a 1 + a 0. The degree of a polynomial is the highest power of in its epression. Constant(non-zero) polynomials, linear polynomials, quadratics, cubics and quartics are polynomials of degree 0, 1, 2,3and4respectively.Thefunction f() = 0isalsoapolynomial,butwesaythatitsdegree is undefined. www.mathcentre.ac.uk 2 c mathcentre 2009
3. Graphs of polynomial functions Wehavemetsomeofthebasicpolynomialsalready.Foreample, f() = 2isaconstantfunction and f() = 2 + 1isalinearfunction. f () f () = 2 + 1 2 f () = 2 1 It is important to notice that the graphs of constant functions and linear functions are always straight lines. Wehavealreadysaidthataquadraticfunctionisapolynomialofdegree2. Herearesome eamples of quadratic functions: f() = 2, f() = 2 2, f() = 5 2. Whatistheimpactofchangingthecoefficientof 2 aswehavedoneintheseeamples? One waytofindoutistosketchthegraphsofthefunctions. f () f () = 5 2 f () = 2 2 f () = 2 Youcanseefromthegraphthat,asthecoefficientof 2 isincreased,thegraphisstretched vertically(that is, in the y direction). Whatwillhappenifthecoefficientisnegative?Thiswillmeanthatallofthepositive f()values willnowbecomenegative.sowhatwillthegraphsofthefunctionslooklike?thefunctionsare now f() = 2, f() = 2 2, f() = 5 2. www.mathcentre.ac.uk 3 c mathcentre 2009
f () f () = 2 f () = 2 2 f () = 5 2 Noticeherethatallofthesegraphshaveactuallybeenreflectedinthe -ais. Thiswillalways happenforfunctionsofanydegreeiftheyaremultipliedby 1. Nowletuslookatsomeotherquadratic functions toseewhathappens whenwevarythe coefficientof,ratherthanthecoefficientof 2.Weshalluseatableofvaluesinordertoplot thegraphs,butweshallfillinonlythosevaluesneartheturningpointsofthefunctions. 5 4 3 2 1 0 1 2 2 + 6 2 0 0 2 6 2 + 4 0 3 4 3 0 2 + 6 5 8 9 8 5 Youcanseethesymmetryineachrowofthetable,demonstratingthatwehaveconcentrated ontheregionaroundtheturningpointofeachfunction. Wecannowusethesevaluestoplot the graphs. f () f () = 2 + f () = 2 + 4 f () = 2 + 6 As you can see, increasing the positive coefficient of in this polynomial moves the graph down andtotheleft. www.mathcentre.ac.uk 4 c mathcentre 2009
What happens if the coefficient of is negative? 2 1 0 1 2 3 4 5 2 6 2 0 0 2 6 2 4 0 3 4 3 0 2 6 5 8 9 8 5 Againwecanusethesetablesofvaluestoplotthegraphsofthefunctions. f () f () = 2 f () = 2 4 f () = 2 6 Asyoucansee,increasingthenegativecoefficientof (inabsoluteterms)movesthegraph downandtotheright. Sonowweknowwhathappenswhenwevarythe 2 coefficient,andwhathappenswhenwe varythe coefficient. Butwhathappenswhenwevarytheconstanttermattheendofour polynomial? Wealreadyknowwhatthegraphofthefunction f() = 2 + lookslike,sohow doesthisdifferfromthegraphofthefunctions f() = 2 + + 1,or f() = 2 + + 5,or f() = 2 + 4?Asusual,atableofvaluesisagoodplacetostart. 2 1 0 1 2 2 + 2 0 0 2 6 2 + + 1 3 1 1 3 7 2 + + 5 7 5 5 7 11 2 + 4 2 4 4 2 2 Ourtableofvaluesisparticularlyeasytocompletesincewecanuseouranswersfromthe 2 + columntofindeverythingelse. Wecanusethesetablesofvaluestoplotthegraphsofthe functions. www.mathcentre.ac.uk 5 c mathcentre 2009
f () f () = 2 + + 5 f () = 2 + + 1 f () = 2 + f () = 2 + 4 Aswecanseestraightaway,varyingtheconstanttermtranslatesthe 2 + curvevertically. Furthermore,thevalueoftheconstantisthepointatwhichthegraphcrossesthe f()ais. 4. Turning points of polynomial functions Aturningpointofafunctionisapointwherethegraphofthefunctionchangesfromsloping downwards to sloping upwards, or vice versa. So the gradient changes from negative to positive, orfrompositivetonegative. Generallyspeaking,curvesofdegree ncanhaveupto (n 1) turning points. For instance, a quadratic has only one turning point. Acubiccouldhaveuptotwoturningpoints,and so would look something like this. However, some cubics have fewer turning points: for eample f() = 3. Butnocubichasmorethantwo turning points. www.mathcentre.ac.uk 6 c mathcentre 2009
Inthesameway,aquarticcouldhaveuptothree turning turning points, and so would look something like this. Again, some quartics have fewer turning points, but none has more. Key Point Apolynomialofdegree ncanhaveupto (n 1)turningpoints. 5. Roots of polynomial functions Youmayrecallthatwhen ( a)( b) = 0,weknowthat aand barerootsofthefunction f() = ( a)( b). Nowwecanusetheconverseofthis,andsaythatif aand bareroots, thenthepolynomialfunctionwiththeserootsmustbe f() = ( a)( b),oramultipleof this. Foreample,ifaquadratichasroots = 3and = 2,thenthefunctionmustbe f() = ( 3)(+2),oraconstantmultipleofthis.Thiscanbeetendedtopolynomialsofanydegree. Foreample,iftherootsofapolynomialare = 1, = 2, = 3, = 4,thenthefunctionmust be f() = ( 1)( 2)( 3)( 4), or a constant multiple of this. Letusalsothinkaboutthefunction f() = ( 2) 2.Wecanseestraightawaythat 2 = 0, sothat = 2. Forthisfunctionwehaveonlyoneroot. Thisiswhatwecallarepeatedroot, andarootcanberepeatedanynumberoftimes. Foreample, f() = ( 2) 3 ( + 4) 4 has arepeatedroot = 2,andanotherrepeatedroot = 4. Wesaythattheroot = 2has multiplicity3,andthattheroot = 4hasmultiplicity4. Theusefulthingaboutknowingthemultiplicityofarootisthatithelpsuswithsketchingthe graphofthefunction.ifthemultiplicityofarootisoddthenthegraphcutsthroughthe -ais atthepoint (, 0). Butifthemultiplicityiseventhenthegraphjusttouchesthe -aisatthe point (, 0). For eample, take the function f() = ( 3) 2 ( + 1) 5 ( 2) 3 ( + 2) 4. Theroot = 3hasmultiplicity2,sothegraphtouchesthe -aisat (3, 0). www.mathcentre.ac.uk 7 c mathcentre 2009
Theroot = 1hasmultiplicity5,sothegraphcrossesthe -aisat ( 1, 0). Theroot = 2hasmultiplicity3,sothegraphcrossesthe -aisat (2, 0). Theroot = 2hasmultiplicity4,sothegraphtouchesthe -aisat ( 2, 0). Totakeanothereample,supposewehavethefunction f() = ( 2) 2 ( + 1). Wecansee thatthelargestpowerof is3,andsothefunctionisacubic. Weknowthepossiblegeneral shapesofacubic,andasthecoefficientof 3 ispositivethecurvemustgenerallyincreaseto therightanddecreasetotheleft.wecanalsoseethattherootsofthefunctionare = 2and = 1. Theroot = 2hasevenmultiplicityandsothecurvejusttouchesthe -aishere, whilst = 1hasoddmultiplicityandsoherethecurvecrossesthe -ais.thismeanswecan sketch the graph as follows. f () 1 2 Key Point Thenumber aisarootofthepolynomialfunction f()if f(a) = 0,andthisoccurswhen ( a) isafactorof f(). If aisarootof f(),andif ( a) m isafactorof f()but ( a) m+1 isnotafactor,thenwe say that the root has multiplicity m. Atarootofoddmultiplicitythegraphofthefunctioncrossesthe -ais,whereasatarootof even multiplicity the graph touches the -ais. Eercises 1. What is a polynomial function? 2. Which of the following functions are polynomial functions? (a) f() = 4 2 + 2 (b) f() = 3 3 2 + (c) f() = 12 4 5 + 3 2 (d) f() = sin + 1 (e) f() = 3 12 2/ (f) f() = 3 11 2 12 www.mathcentre.ac.uk 8 c mathcentre 2009
3. Write down one eample of each of the following types of polynomial function: (a) cubic (b) linear (c) quartic (d) quadratic 4.Sketchthegraphsofthefollowingfunctionsonthesameaes: (a) f() = 2 (b) f() = 4 2 (c) f() = 2 (d) f() = 4 2 5.Considerafunctionoftheform f() = 2 +a,where arepresentsarealnumber.thegraph of this function is represented by a parabola. (a)when a > 0,whathappenstotheparabolaas aincreases? (b)when a < 0,whathappenstotheparabolaas adecreases? 6.Writedownthemaimumnumberofturningpointsonthegraphofapolynomialfunctionof degree: (a) 2 (b) 3 (c) 12 (d) n 7. Write down a polynomial function with roots: (a) 1, 2, 3, 4 (b) 2, 4 (c) 12, 1, 6 8. Write down the roots and identify their multiplicity for each of the following functions: (a) f() = ( 2) 3 ( + 4) 4 (b) f() = ( 1)( + 2) 2 ( 4) 3 9. Sketch the following functions: (a) f() = ( 2) 2 ( + 1) (b) f() = ( 1) 2 ( + 3) Answers 1.Apolynomialfunctionisafunctionthatcanbewrittenintheform f() = a n n + a n 1 n 1 + a n 2 n 2 +... + a 2 2 + a 1 + a 0, whereeach a 0, a 1,etc.representsarealnumber,andwhere nisanaturalnumber(including0). 2. (a) f() = 4 2 + 2isapolynomial (b) f() = 3 3 2 + isnotapolynomial,becauseof (c) f() = 12 4 5 + 3 2 isapolynomial (d) f() = sin + 1isnotapolynomial,becauseof sin (e) f() = 3 12 2/isnotapolynomial,becauseof 2/ (f) f() = 3 11 2 12 isapolynomial 3. (a) Thehighestpowerof mustbe3,soeamplesmightbe f() = 3 2 + 1or f() = 3 2. (b) Thehighestpowerof mustbe1,soeamplesmightbe f() = or f() = 6 5. (c) Thehighestpowerof mustbe4,soeamplesmightbe f() = 4 3 3 + 2or f() = 4 5. (d) Thehighestpowerof mustbe2,soeamplesmightbe f() = 2 or f() = 2 5. 4. www.mathcentre.ac.uk 9 c mathcentre 2009
f () f () = 4 2 f () = 2 f () = 2 f () = 4 2 5. (a) When a > 0,theparabolamovesdownandtotheleftas aincreases. (b) When a < 0,theparabolamovesdownandtotherightas adecreases. 6. (a) 1 turning point (d) (n 1) turning points. (b) 2 turning points (c) 11 turning points 7. (a) f() = ( 1)( 2)( 3)( 4)oramultiple (b) f() = ( 2)( + 4)oramultiple (c) f() = ( 12)( + 1)( + 6)oramultiple. 8. (a) = 2 = 4 (b) = 1 = 2 = 4 odd multiplicity even multiplicity odd multiplicity even multiplicity odd multiplicity www.mathcentre.ac.uk 10 c mathcentre 2009
9) f () f () = ( 1) 2 ( + 3) f () = ( 2) 2 ( + 1) www.mathcentre.ac.uk 11 c mathcentre 2009