write:Also recallthatx-interceptscaneithercrossthex-axisortheycanjusttouchthex-axiswithoutactuallycrossingtheaxis. Noticeaswellfromthegraphsabovethatthex-interceptscaneitherflattenoutast

P(x)
providedP(r)=0.Butthismeansthatx=risalsoasolutiontoP(x)=0.Inotherwords
thezeroesofapolynomialarealsothex-interceptsofthegraph.Also
recallthatx-interceptscaneithercrossthex-axisortheycanjusttouchthex-axiswithoutactuallycrossingtheaxis.Noticeaswellfromthegraphsabovethatthex-interceptscaneitherflattenoutastheycrossthex-axisortheycangothroughthex-axisatanangle.Thefollowingfactwillrelatealloftheseideastothemultiplicityofthezero.FactIfx=risazeroofthepolynomialP(x)withmultiplicitykthen
1.Ifkisoddthenthex-interceptcorrespondingtox=rwillcrossthex-axis.2.Ifkiseventhenthex-interceptcorrespondingtox=rwillonlytouchthex-axisandnotactuallycrossit.Furthermore
ifk>1thenthegraphwillflattenoutatx=r.Finally
noticethatasweletxgetlargeinboththepositiveornegativesense(i.e.ateitherendofthegraph)thenthegraphwilleitherincreasewithoutboundordecreasewithoutbound.Thiswillalwayshappenwitheverypolynomialandwecanusethefollowingtesttodeterminejustwhatwillhappenattheendpointsofthegraph.LeadingCoefficientTestSupposethatP(x)isapolynomialwithdegreen.Soweknowthatthepolynomialmustlooklike
P(x)=axn+


Wedon’tknowifthereareanyothertermsinthepolynomial
butwedoknowthatthefirsttermwillhavetobetheonelistedsinceithasdegreen.WenowhavethefollowingfactsaboutthegraphofP(x)attheendsofthegraph.©PaulDawkinsAlgebra–293–