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45:EvenMoreBinomial Expansions?ChristineCrisp“TeachALevelMaths”
Vol.2:A2CoreModules
ModuleC4Certainimagesand/orphotosonthispresentationarethecopyrightedpropertyofJupiterImagesandarebeingusedwithpermissionunderlicense.Theseimagesand/orphotosmaynotbecopiedordownloadedwithoutpermissionfromJupiterImages
Thebinomialexpansioncanalsobeusedwithalgebraicfractions.Solution:WejustwritetheexpressionasOnlythe2ndfactorneedsexpanding.e.g.Findtheexpansioninascendingpowersofx,uptoandincludingtheterm,forthefollowingandgivethevaluesofxforwhichtheexpansionisvalid.
So,Replacenbyandxby(3x)
StartwithWedon’tneedanytermswithpowershigherthan.
thenStartwithWedon’tneedanytermswithpowershigherthan.
thenStartwithfinallyWedon’tneedanytermswithpowershigherthan.
thenStartwithfinallyWedon’tneedanytermswithpowershigherthan.
So,ValidforWedon’tneedanytermswithpowershigherthan.
ExercisesFindthe1st4terms,intheexpansioninascendingpowersofx,forthefollowingandgivethevaluesofxforwhichtheexpansionisvalid.
Solution:Replacenbyandxby(-x)
Solution:So,Validfor
Theexpressionswe’veusedinbinomialexpansionshaveallbeenoftheformThe1sttermis1,andthisistheONLYformwecanuse.Anyotherexpressionswewanttoexpandmustbewrittenwith1asthe1stterm.Toexpandforexamplewemustremove4fromthebrackets.
WewriteWhilstthe4isinsidethebracketsitisincludedunderthepowerso,whenwetakethe4outside,acopyofthepowermustcomewithit.
WewriteThe4outsidethebracketsmultipliesbothtermsinside,sothe2ndtermmustbedividedby4.
WewriteWenowexpandasbefore,rememberingtomultiplyby2.
Thereisn’taconvenientplacetoputthe2!soIwriteitastoavoidpilingupthefractions.ReplacenbyandxbyValidfor
SUMMARYToexpande
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