Polynomial functions - Math Centre

1. IntroductionApolynomialfunctionisafunctionsuchasaquadratic,acubic,aquartic,andsoon,involvingonlynon-negativeintegerpowersof x. Wecangiveageneraldefintionofapolynomial,anddefineitsdegree.2. What is a polynomial?Apolynomialofdegree nisafunctionoftheformf(x) = a n x n + a n−1 x n−1 + . . . + a 2 x 2 + a 1 x + a 0wherethe a’sarerealnumbers(sometimescalledthecoefficientsofthepolynomial).Althoughthisgeneralformulamightlookquitecomplicated,particularexamplesaremuchsimpler. Forexample,f(x) = 4x 3 − 3x 2 + 2isapolynomialofdegree3,as3isthehighestpowerof xintheformula.Thisiscalledacubicpolynomial,orjustacubic.Andf(x) = x 7 − 4x 5 + 1isapolynomialofdegree7,as7isthehighestpowerof x.Noticeherethatwedon’tneedeverypowerof xupto7: weneedtoknowonlythehighestpowerof xtofindoutthedegree. Anexampleofakindyoumaybefamiliarwithisf(x) = 4x 2 − 2x − 4whichisapolynomialofdegree2,as2isthehighestpowerof x.Thisiscalledaquadratic.Functionscontainingotheroperations,suchassquareroots,arenotpolynomials.Forexample,f(x) = 4x 3 + √ x − 1isnotapolynomialasitcontainsasquareroot.Andf(x) = 5x 4 − 2x 2 + 3/xisnotapolynomialasitcontainsa‘divideby x’.ApolynomialisafunctionoftheformKey Pointf(x) = a n x n + a n−1 x n−1 + . . . + a 2 x 2 + a 1 x + a 0 .Thedegreeofapolynomialisthehighestpowerof xinitsexpression. Constant(non-zero)polynomials,linearpolynomials,quadratics,cubicsandquarticsarepolynomialsofdegree0,1,2,3and4respectively.Thefunction f(x) = 0isalsoapolynomial,butwesaythatitsdegreeis‘undefined’.www.mathcentre.ac.uk 2 c○mathcentre2009