The Binomial Expansion 二次项展开式

TheBinomialExpansionIFYMaths1TheBinomialExpansionLearningOutcomes1.Expandforsmallpositiveintegern2.UsePascal’striangletofindthebinomialcoefficients3.Expandforsmallpositiveintegernnba)(nx)1(TheBinomialExpansionPowersofa+bInthispresentationwewilldevelopaformulatoenableustofindthetermsoftheexpansionofnba)(wherenisanypositiveinteger.Wecalltheexpansionbinomialastheoriginalexpressionhas2parts.TheBinomialExpansionPowersofa+b222baba2)(ba))((babaWeknowthatsothecoefficientsofthetermsare1,2and1Wecanwritethisas22baba121TheBinomialExpansion2ab1ba22)2)((22bababaPowersofa+b3)(ba2))((baba3a1TheBinomialExpansion2ab23b1223abbaa121)2)((22bababaPowersofa+b3)(ba2))((bababa21TheBinomialExpansion223abbaa121)2)((22bababaPowersofa+b3)(ba2))((baba322babba1213223babbaa3311TheBinomialExpansion223abbaaPowersofa+b3)(ba2))((baba)2)((22bababa322babba3223babbaasothecoefficientsoftheexpansionofare1,3,3and13)(ba1211213311TheBinomialExpansionPowersofa+b4)(ba3))((baba)33)((3223babbaaba32234abbabaa133143223babbaba1331432234babbabaa64141TheBinomialExpansion32234abbabaaPowersofa+b4)(ba3))((baba)33)((3223babbaaba43223babbaba432234babbabaa1331336414111Thiscoefficient......isfoundbyadding3and1;thecoefficientsthatarein3)(baTheBinomialExpansion31432234abbabaaPowersofa+b4)(ba3))((baba)33)((3223babbaaba43223babbaba432234babbabaa1333614111Thiscoefficient......isfoundbyadding3and1;thecoefficientsthatarein3)(baTheBinomialExpansionPowersofa+bSo,wenowhave3)(ba2)(baCoefficientsExpression12113314)(ba14641TheBinomialExpansionSo,wenowhave3)(ba2)(baCoefficientsExpression12113314)(ba14641Eachnumberinarowcanbefoundbyaddingthe2coefficientsaboveit.Powersofa+bTheBinomialExpansionPowersofa+bSo,wenowhave3)(ba2)(baCoefficientsExpression12113314)(ba14641The1standlastnumbersarealways1.Eachnumberinarowcanbefoundbyaddingthe2coefficientsaboveit.TheBinomialExpansionPowersofa+bSo,wenowhave3)(ba2)(baCoefficientsExpression12113311)(ba110)(ba4)(ba14641Tomakeatriangleofcoefficients,wecanfillintheobviousonesatthetop.1TheBinomialExpansionPowersofa+bThetriangleofbinomialcoefficientsiscalledPascal’striangle,aftertheFrenchmathematician....butit’seasytoknowwhichrowwewantas,forexample,3)(bastartswith13...10)(bawillstart110...Noticethatthe4throwgivesthecoefficientsof)(ba3TheBinomialExpansionExerciseFindthecoefficientsintheexpansionof6)(baTheBinomialExpansionWeusuallywanttoknowthecompleteexpansionnotjustthecoefficients.Powersofa+b5)(bae.g.FindtheexpansionofPascal’strianglegivesthecoefficientsSolution:15101105ThefullexpansionisTip:Thepowersineachtermsumto554322345babbababaa151010511TheBinomialExpansione.g.2Writeouttheexpansionofinascendingpowersofx.4)1(xPowersofa+bSolution:Thecoefficientsarea4322344464)(aaaabbbbbTogetweneedtoreplaceaby14)1(x(Ascendingpowersjustmeansthatthe1sttermmusthavethelowestpowerofxandthenthepowersmustincrease.)14614WeknowthatTheBinomialExpansion14322344464)(1(1)(1)(1)bbbbbe.g.2Writeouttheexpansionofinascendingpowersofx.14614WeknowthatPowersofa+bSolution:ThecoefficientsareTogetweneedtoreplaceaby14)1(x4)1(xTheBinomialExpansion4322344464)(Becareful!Theminussign...issquaredaswellasthex.Thebracketsarevital,otherwisethesignswillbewrong!e.g.2Writeouttheexpansionofinascendingpowersofx.14614WeknowthatPowersofa+bSolution:ThecoefficientsareTogetweneedtoreplaceaby1andbby(-x)4)1(x1(1)1(1)(1)(-x)(-x)(-x)(-x)(-x)Simplifyinggives4)1(x1x426x34x4x4)1(xTheBinomialExpansion4)1(xTogetweneedtoreplaceaby1andbby(-x)Sinceweknowthatanypowerof1equals1,wecouldhavewritten1here...e.g.2Writeouttheexpansionofinascendingpowersofx.14614WeknowthatPowersofa+bSolution:Thecoefficientsare4322344464)(11(1)(1)(1)(-x)(-x)(-x)(-x)(-x)Simplifyinggives4)1(x1x426x34x4x4)1(xTheBinomialExpansion4)1(xTogetweneedtoreplaceaby1andbby(-x)e.g.2Writeouttheexpansionofinascendingpowersofx.14614WeknowthatPowersofa+bSolution:Thecoefficientsare432234464)(11(1)(1)(1)(-x)(-x)(-x)(-x)(-x)Simplifyinggives4)1(x1x426x34x4xSinceweknowthatanypowerof1equals1,wecouldhavewritten1here...4)1(xTheBinomialExpansion4)1(xTogetweneedtoreplaceaby1andbby(-x)e.g.2Writeouttheexpansionofinascendingpowersofx.14614WeknowthatPowersofa+bSolution:Thecoefficientsare432234464)(11(1)(1)(1)(-x)(-x)(-x)(-x)(-x)Simplifyinggives4)1(x1x426x34x4x...andmissedthese1sout.4)1(xTheBinomialExpansione.g.2Writeouttheexpansionofinascendingpowersofx.14614WecouldgostraighttoPowersofa+bSolution:Thecoefficientsare4324464)(11(-x)(-x)(-x)(-x)(-x)Simplifyinggives4)1(x1x426x34x4x4)1(xTheBinomialExpansionExercise1.Findtheexpansionofinascendingpowersofx.5)21(xTheBinomialExpansionPowersofa+b20)(baIfwewantthefirstfewtermsoftheexpansionof,forexample,,Pascal’striangleisnothelpful.Wewillnowdevelopamethodofgettingthecoefficientswithoutneedingthetriangle.TheBinomialExpansionEachcoefficientcanbefoundbymultiplyingthepreviousonebyafraction.Thefract