Polynomial functions - Math Centre

•Theroot x = −1hasmultiplicity5,sothegraphcrossesthe x-axisat (−1, 0).•Theroot x = 2hasmultiplicity3,sothegraphcrossesthe x-axisat (2, 0).•Theroot x = −2hasmultiplicity4,sothegraphtouchesthe x-axisat (−2, 0).Totakeanotherexample,supposewehavethefunction f(x) = (x − 2) 2 (x + 1). Wecanseethatthelargestpowerof xis3,andsothefunctionisacubic. Weknowthepossiblegeneralshapesofacubic,andasthecoefficientof x 3 ispositivethecurvemustgenerallyincreasetotherightanddecreasetotheleft.Wecanalsoseethattherootsofthefunctionare x = 2andx = −1. Theroot x = 2hasevenmultiplicityandsothecurvejusttouchesthe x-axishere,whilst x = −1hasoddmultiplicityandsoherethecurvecrossesthe x-axis.Thismeanswecansketchthegraphasfollows.f (x)−12xKey PointThenumber aisarootofthepolynomialfunction f(x)if f(a) = 0,andthisoccurswhen (x−a)isafactorof f(x).If aisarootof f(x),andif (x − a) m isafactorof f(x)but (x − a) m+1 isnotafactor,thenwesaythattheroothasmultiplicity m.Atarootofoddmultiplicitythegraphofthefunctioncrossesthe x-axis,whereasatarootofevenmultiplicitythegraphtouchesthe x-axis.Exercises1.Whatisapolynomialfunction?2.Whichofthefollowingfunctionsarepolynomialfunctions?(a) f(x) = 4x 2 + 2 (b) f(x) = 3x 3 − 2x + √ x (c) f(x) = 12 − 4x 5 + 3x 2(d) f(x) = sin x + 1 (e) f(x) = 3x 12 − 2/x (f) f(x) = 3x 11 − 2x 12www.mathcentre.ac.uk 8 c○mathcentre2009