CfE Advanced Higher Mathematics Course materials Topic 5: Binomial theorem

Transcription

1 SCHOLAR Study Guide CfE Advaced Highe Mathematics Couse mateials Topic : Biomial theoem Authoed by: Fioa Withey Stilig High School Kae Withey Stilig High School Reviewed by: Magaet Feguso Peviously authoed by: Jae S Pateso Doothy A Watso Heiot-Watt Uivesity Edibugh EH AS, Uited Kigdom.

2 Fist published 06 by Heiot-Watt Uivesity. This editio published i 0 by Heiot-Watt Uivesity SCHOLAR. Copyight 0 SCHOLAR Foum. Membes of the SCHOLAR Foum may epoduce this publicatio i whole o i pat fo educatioal puposes withi thei establishmet povidig that o pofit accues at ay stage, Ay othe use of the mateials is goveed by the geeal copyight statemet that follows. All ights eseved. No pat of this publicatio may be epoduced, stoed i a etieval system o tasmitted i ay fom o by ay meas, without witte pemissio fom the publishe. Heiot-Watt Uivesity accepts o esposibility o liability whatsoeve with egad to the ifomatio cotaied i this study guide. Distibuted by the SCHOLAR Foum. SCHOLAR Study Guide Couse Mateials: CfE Advaced Highe Mathematics. CfE Advaced Highe Mathematics Couse Code: C

3 Ackowledgemets Thaks ae due to the membes of Heiot-Watt Uivesity's SCHOLAR team who plaed ad ceated these mateials, ad to the may colleagues who eviewed the cotet. We would like to ackowledge the assistace of the educatio authoities, colleges, teaches ad studets who cotibuted to the SCHOLAR pogamme ad who evaluated these mateials. Gateful ackowledgemet is made fo pemissio to use the followig mateial i the SCHOLAR pogamme: The Scottish Qualificatios Authoity fo pemissio to use Past Papes assessmets. The Scottish Govemet fo fiacial suppot. The cotet of this Study Guide is aliged to the Scottish Qualificatios Authoity SQA cuiculum. All bad ames, poduct ames, logos ad elated devices ae used fo idetificatio puposes oly ad ae tademaks, egisteed tademaks o sevice maks of thei espective holdes.

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5 Topic Biomial theoem Cotets. Lookig back Revisio Factoials Biomial coefficiets Pascal's tiagle Biomial theoem Fidig coefficiets Sigma otatio ad biomial theoem applicatios Leaig poits Poofs Exteded ifomatio Ed of topic test

6 TOPIC. BINOMIAL THEOREM Leaig objective By the ed of this topic, you should be able to: udestad the otatio!; evaluate! fo a value of N; ewite! i tems of factoials smalle tha ; idetify what the biomial coefficiet is ad evaluate it; use the diffeet otatios fo the biomial coefficiet; detemie elatioships betwee biomial coefficiets; costuct Pascal's tiagle; udestad the coectio betwee Pascal's tiagle ad the biomial coefficiets; use the biomial theoem to expad a expessio of the fom x + y whee x, y Q ad N; wite dow a expessio fo the geeal tem i x + y ; fid the coefficiet of a paticula tem i a give expessio of the fom ax + by o ax + b y ; use the biomial theoem to evaluate a decimal to a powe e.g. 0.

7 TOPIC. BINOMIAL THEOREM. Lookig back Summay of pio kowledge Expadig double backet expessios: Natioal Whe expadig double backets, emembe that each tem i the secod backet is multiplied by each tem i the fist backet by tuig the expessio ito two sigle backet expessios o usig the aibow method o FOIL. Whe addig ad subtactig factios: Natioal Detemie a commo deomiato by multiplyig the deomiatos togethe o fidig the lowest commo multiple of the deomiatos. Chage each factio to a equivalet factio usig the commo deomiato, the add/subtact the umeatos - lastly, simplify the esultig factio if possible. Simplifyig factios: Natioal To simplify factios, factoise the umeato ad deomiato whee possible, the cacel commo factos of the umeato ad deomiato. Multiplyig factios: Natioal To multiply factios, multiply the umeatos togethe ad multiply the deomiatos togethe. Simplify the esultig factio if possible. Dividig factios: Natioal Whe dividig by a factio, take the diviso the factio that you ae dividig by ad tu it upside dow. At the same time, chage the divisio to a multiplicatio. Now multiply the factios ad simplify the esultig factio if possible.

8 TOPIC. BINOMIAL THEOREM.. Revisio Expadig backets Example Poblem: Expad ad simplify y + y Solutio: To multiply out this set of backets we multiply two backets out ad get a aswe. We the take this aswe ad multiply it by the emaiig backet. Remembe that to multiply out backets we take each tem i oe backet ad multiply it by each tem i the secod backet. Whe thee ae oly two tems i each backet this method is kow as the FOIL method Fist Oute Ie Last. Witte out i full: y + y y + y y Multiply out y y fist: y + y y + y 0y + Multiply each tem i the fist backet by each tem i the secod backet, the simplify: y +y y y 0y + + y 0y + y 60y +y + 8y 0y +0 y y +y +0 Factoisig ito backets Examples. Poblem: Factoise y + y + Solutio: We wat two tems which multiply to make y. y y y so ou backets stat y y. Now we wat two umbes which multiply to make +, i.e. o. I this example we eed to check that the sum of the poducts of the ie ad oute tems gives us the middle tem y.

9 TOPIC. BINOMIAL THEOREM Multiplyig the ie tems gives y ad multiplyig the oute tems gives y sum y, but the middle tem we wat is y. Multiplyig the ie tems gives y ad multiplyig the oute tems y sum y. This gives us the middle tem that we wat. So we have: y +y +y +y +. Poblem: Factoise g 0g +6 Solutio: We should always check fo a simple commo facto fist. This questio has a commo facto of givig g 0g + Next we wat two tems to make g, g g g, so ou backets stat g g Now we wat two umbes which multiply to make +, i.e. o. I this example we eed to check that the sum of the poducts of the ie ad oute tems gives the middle tem 0g. g + g +makes the poduct of the ie tems g ad the oute tems g sum 0g; we ealy have it, but we wated 0g. The sig is wog, but we kow that also makes, sowehave: g g g 0g + Hece: g 0g +6 g 0g + g g Addig factios Example Poblem: Add ad togethe. Solutio: +? we eed a commo deomiato + multiply fo a commo deomiato ow we ca add the factios togethe 0 0 chage the impope factio to a mixed umbe

10 6 TOPIC. BINOMIAL THEOREM Multiplyig factios Example Poblem: 6 Solutio: To multiply factios: multiply the umeatos togethe; multiply the deomiatos togethe; simplify the esultig factio Divisio Example Poblem: Without a calculato divide 60 by usig log divisio. Solutio: does ot go ito o, but it goes ito 6 eight times. Multiply by 8 to give 0, wite that udeeath 6 ad the subtact. Big the 0 of 60 dow beside the 6 to make 60. goes ito 60 fou times.

11 TOPIC. BINOMIAL THEOREM Multiply by to give 60, wite that udeeath the 60 aleady thee ad the subtact. The aswe is 0 which meas thee is o emaide. Theefoe, 60 8 The followig evisio pactice ad execise should help to idetify ay aeas of weakess i techiques which ae equied fo the study of this topic. Some evisio may be ecessay if ay of the questios seem difficult. Revisio pactice Go olie Q: Expad ad simplify y y + Q: Expad ad simplify y Q: Factoise g + g + Q: Factoise h +h Q: Factoise j +j Q6: Factoise k k Q: + Q8: 8 Q9: Q0: 6 8 Q: Without a calculato, divide 68 by 8 usig log divisio. Q: Without a calculato, divide 0 by usig log divisio.

12 8 TOPIC. BINOMIAL THEOREM. Factoials The Biomial Theoem is a useful tool i may baches of mathematics. Oe impotace is that is gives a efficiet way of coutig combiatios. Fo istace, with the Biomial Theoem you ca quickly calculate the umbe of distict ways of choosig six umbes fo you lottey ticket. As you pogess though the topics you will see the Biomial Theoem appea i othe seemigly uelated topics. I ode to use the Biomial Theoem we eed to udestad what a factoial is, how to evaluate ad maipulate it. The followig sectio will cove this. The defiitio of!! called factoial is the poduct of the iteges,,,...,,, i.e.!... fo N Examples.!...!!! 0 Factoial value Poblem: What is the value of 6!? Solutio: 6! 6 0 Q: What is!? Q: What is!? Note that as iceases,! apidly iceases. Key poit!...

13 TOPIC. BINOMIAL THEOREM 9 Factoials calculato activity Fid the lagest value of that ca be eteed i a calculato as! without givig a eo message. What is the value of this factoial? Is this a accuate aswe? Factoial fomula Give that! is the poduct of the iteges,,,...,, fo N ad! is the poduct of the iteges,,...,,, it follows that:!...!...!! so!! Examples. Poblem: What is! i tems of 6!? Solutio: Usig!!:! 6! 6! 6!. Poblem: What othe factoials could! be witte i tems of? Solutio: Usig!!:! 6! 6! o! 6!! o! 6! 0!

14 0 TOPIC. BINOMIAL THEOREM ad so o.... Poblem: What is 9! i tems of 8!? Solutio: Usig!!: 9! ! ! 9 8! Factoials pactice Go olie Q: What is 8! i tems of!? Q6: What is! i tems of!? Q: Wite 8! i tems of!? Q8: Wite 0! i tems of 8!? Q9: Wite! i tems of 9!? Q0: Wite 6! i tems of!?! has bee defied fo N as! 0! By covetio, 0! is give the value. Key poit Zeo factoial has a value of. That is, 0! Note that this still fits the ule!! because! ad 0!

15 TOPIC. BINOMIAL THEOREM A video explaiig why 0! Go olie Please watch the followig YouTube video which explais why 0! : Regadig the equatio at the ed: James says it should be e -t dt NOT e - d... soy fo the mix-up!

16 TOPIC. BINOMIAL THEOREM. Biomial coefficiets! Fo iteges N ad 0, the umbe give by!! is called a Biomial Coefficiet. It is deoted by Key poit The biomial coefficiet fomula is:!!! Example : Biomial coefficiet Poblem: Evaluate usig the biomial coefficiet Solutio:!!!!!! 6 6!!! The biomial coefficiet is also deoted C ad is used i aothe elated maths topic called combiatoics. Combiatoics is ot coveed i this couse, but the tem idicates the umbe of ways of choosig elemets fom a set of elemets. Example : C Poblem: How may ways ca two chocolates be chose fom a box cotaiig twety chocolates?

17 TOPIC. BINOMIAL THEOREM Solutio: 0 0 C 0!!8! 0 9 8! 8! That is, two chocolates be chose fom a box cotaiig twety chocolates 90 ways. Biomial coefficiets: C pactice Go olie Q: What is? 6 Q: What is? Q: What is? Q: How may ways ca fou pupils be chose fom a goup of seve? Fist ule of biomial coefficiets Fom the biomial coefficiet!!!!!! 0!!! otice the symmety!!!!!! 0 Notice that 0 ad 0 ad so

18 TOPIC. BINOMIAL THEOREM This illustates the fist ule fo biomial coefficiets. Key poit The fist biomial coefficiet ule is: The poof is ot equied fo exam puposes, but is available simply fo you iteest. Poof : Fist biomial coefficiet ule Poof Pove that, the same as povig that!!!!! +!!!!!!! Fist biomial coefficiet ule pactice Go olie Q: Fid aothe biomial coefficiet equal to Q6: Fid aothe biomial coefficiet equal to Secod ule of biomial coefficiets 6 Fom the ealie questios 0, ad

19 TOPIC. BINOMIAL THEOREM 6 Theefoe + This illustates the secod ule fo biomial coefficiets. Key poit The secod biomial coefficiet ule is: + + The poof is ot equied fo exam puposes, but is available simply fo you iteest. Poof : Secod biomial coefficiet ule Poof + Pove that + I this poof, the followig facts ae equied:.!!. +! +!. Two factios ca be combied ove a commo deomiato. + +!!! +!!!!! +! + +!! +!! + +!! +!! + +! +!! +! +! +!! +! +

20 6 TOPIC. BINOMIAL THEOREM Secod biomial coefficiet ule pactice Go olie 8 8 Q: Wite dow + as a biomial coefficiet. 6 Q8: Wite dow + as a biomial coefficiet. Example : Fidig give the value of a biomial coefficiet ad Poblem: If is a positive itege such that, fid. Solutio: Fom the biomial coefficiet:!!! Now simplify by cacellig factos: Multiplyig out ad simplifyig: 0 To solve a quadatic, we eed to factoise: o Sice is a membe of the atual umbes N, the 6 Fidig give the value of a biomial coefficiet ad pactice Go olie Q9: Fid the positive itege such that 0 Q0: Fid the positive itege such that Q: Fid the positive itege such that 6

21 TOPIC. BINOMIAL THEOREM Biomial coefficiets pactice Go olie Q: Evaluate 0! Q: Evaluate! Q: What is 00! as a poduct of a itege ad the factoial of 99? Q: What is! as a poduct of a itege ad the factoial of 0? Q6: Evaluate 6 C 9 Q: Evaluate Q8: Evaluate Q9: Fid aothe biomial coefficiet equal to 6 Q0: Fid aothe biomial coefficiet equal to Q: Wite dow + as a biomial coefficiet. Q: Wite dow + as a biomial coefficiet. 9 0 Biomial coefficiets execise Go olie Q: Evaluate! Q: What is! as a poduct of a itege ad a factoial?

22 8 TOPIC. BINOMIAL THEOREM 8 Q: Evaluate Q6: Fid aothe biomial coefficiet equal to Q: Wite dow + as a biomial coefficiet.. Pascal's tiagle Blaise Pascal was cedited with discoveig what is kow as 'Pascal's tiagle'. It is made up of iteges set out as a tiagle. The umbe appeas at the top ad at each ed of subsequet ows. The umbes i the body of the tiagle follow the ule: To fid a umbe, add the two umbes fom the above left ad above ight of it. The ows ae umbeed fom Row 0. Pascal's tiagle Go olie Q8: Complete ows to of Pascal's tiagle. Row 0 Row Row Row Row Row Row 6 Row

23 TOPIC. BINOMIAL THEOREM 9 Q9: The followig is a table of the same desig as Pascal's tiagle, oly this time the eties ae the biomial coefficiets as show. Row 0 Row Row Row What ae the coespodig values of these biomial coefficiets? Remembe that!!! Q0: Complete ows to with biomial coefficiets ad thei coespodig values. Row Row Row 6 Row Q: What is the coectio betwee Pascal's tiagle ad the biomial coefficiets table? Q: What is the biomial coefficiet fo the fouth tem i ow? Q: What is the biomial coefficiet fo the thid tem i ow? Q: What would be the biomial coefficiet fo the seveth tem i ow 0? Q: What would be the biomial coefficiet fo the fifth tem i ow?

24 0 TOPIC. BINOMIAL THEOREM Now if ow 9 is equied, istead of witig out Pascal's tiagle, it is simply a matte of takig the biomial coefficiets of ow 9, amely: ,,,,,,,, ad 9 9 Eithe method ca be used. The eties i Pascal's tiagle ad the coespodig biomial coefficiets ae equal. + Recall that the secod ule fo biomial coefficiets is + This is the defiitio of how to costuct the eties i Pascal's tiagle. If the umbes i the th ow of Pascal's tiagle ae the biomial coefficiets, the the th ety i the ext ow, ow +, is the sum of the th ety ad the th ety i ow. The th ety i the + th ow is + Hece, the eties i the + th ow of Pascal's tiagle ae also the biomial coefficiets. A example of this computatio follows: Row Row sum... Row Row Key poit + +

25 TOPIC. BINOMIAL THEOREM Pascal's tiagle pactice Go olie Q6: What is the biomial coefficiet equivalet to +? Q: What is the biomial coefficiet equivalet to +? 6 Q8: What is the biomial coefficiet equivalet to? 6 Q9: What is the biomial coefficiet equivalet to?. Biomial theoem Look at the followig expasios of x + y fo 0,,, Row 0 x + y 0 Row x + y x + y Row x + y x +xy + y Row x + y x +x y +xy + y The coefficiets still follow the same patte as Pascal's tiagle ad the equivalet table of biomial coefficiets. Coefficiets Row 0 Row x + y Row x + xy + y Row x + x y + xy + y O equivaletly, sice Pascal's tiagle ad the tiagle of Biomial coefficiets ae the same:

26 TOPIC. BINOMIAL THEOREM Biomial coefficiets Row Row x y 0 0 Row x xy y 0 0 Row x x y xy y 0 0 Featues of Biomial expasio Look caefully at the expasio of x + y : x + y x +x y +6x y +xy + y Notice that the x tems stat with a powe of which deceases by i successive tems. Notice that the y tems stat with a powe of 0 which iceases by i successive tems. Notice that fo each set of tems, the powes of x ad y add up to. Usig these obsevatios, we ca e-wite the expasio of x + y i a diffeet way: Fo example: x y ca be witte as x - y ad y ca be witte as x - y This is a example of the biomial theoem ad leads to two defiitios: oe fo the expasio, ad oe fo ay tem withi the expasio. Key poit The biomial theoem states that if x, y R ad N the: x + y x + x y + x y x y y

27 TOPIC. BINOMIAL THEOREM Key poit The geeal tem of x + y is give by: x y The poof is ot equied fo exam puposes, but is available simply fo you iteest. Poof : Biomial theoem Poof Pove the biomial theoem which states that if x, y R ad, N, the: x + y x + x y + x y x y + 0 x y 0 This poof is by iductio, the method of which will be coveed late i the couse. y Let, the: LHS x + y x + y x + y RHS x y x + y x 0 y So LHS RHS ad the theoem holds fo Now, suppose that the esult is tue fo k whee k, the: k k k k k x + y k x k + x k y + x k y xy k + 0 k k Coside: x + y k+ x + yx + y k xx + y k + yx + y k k k k k k x k+ + x k y + x k y x y k + xy k + 0 k k k k k k k x k y + x k y + x k y xy k + 0 k k y k y k+

28 TOPIC. BINOMIAL THEOREM x + y k+ [ ] [ ] k k k k k x k+ + + x k y + + x k y [ ] [ ] k k k k k + x y k + + xy k + k k k k k k + k + x k+ + x k y xy k +y k+ k k+ k + x k+ y 0 Hece, if it is tue fo k, it is also tue fo k +. Howeve, it was also tue fo so it is tue fo all values of. k k k + Remembe that fo all ad, ad that + 0 j + j j See the secod biomial coefficiet ule. y k+ Example Poblem: Use the biomial theoem to expad x + y Solutio: I this case So we have: x + y x + x y + x y x y At this stage we ca evaluate coefficiets usig: biomial coefficiets, e.g.!!!! 0 0! 0!! 0!! ad so o... y

29 TOPIC. BINOMIAL THEOREM Pascal's tiagle x + y x + x y + x y + x y + xy + 0 x +x y +0x y +0x y +xy + y y Biomial expasio Go olie Example Poblem: Usig the biomial theoem, expad x + y x + y 0 x + x y + x y x y Solutio: x + y x + x 6 y + x y x y + xy x +x 6 y +x y +x y +x y +x y +xy 6 + y y y Remembe that: the Biomial coefficiets take the fom: powe that x + y has bee aised to; 0,,... whee is the fo the expasio x + y, sice x is the fist tem i the backet its powe will decease fom, ad sice y is the secod tem i the backet to be expaded its powes will icease to, e.g. x + y x +x y +6x y + xy + y

30 6 TOPIC. BINOMIAL THEOREM Example x + y 8 8 x x 8 y + x 6 y 8 + x y 8 + x y x y 8 + x y xy 8 + y x 8 +8x y +8x 6 y +6x y +0x y +6x y +8x y 6 +8xy + y 8 The biomial theoem woks also with multiples of x ad y, ad othe symbols such as a, b o α, β. Biomial theoem execise Go olie Q60: Use the biomial theoem to expad x + y 6 Q6: Use the biomial theoem to expadx + y 9 Q6: Use the biomial theoem to expad x + y Q6: Use the biomial theoem to expad x + y Q6: Use the biomial theoem to expad x + y 8 Example : Biomial expasio Poblem: Use the biomial theoem to expad x y Solutio: Fom the biomial theoem: x y x + x y+ x y + 0 x y + y I this case, x is eplaced by x ad y is eplaced by y Now we ca evaluate the biomial coefficiets usig Pascal's tiagle: x y x +x y + 6x y + x y + y

31 TOPIC. BINOMIAL THEOREM Evaluate the powes of x ad y: x y 6x + 8x y + 6 x 9y + x y +8y Simplify each tem by multiplyig the coefficiets togethe: x y 6x 96x y + 6x y xy +8y Futhe biomial expasio Go olie Example Poblem: Usig the biomial theoem, expad x +y x + y 0 x + x y + x y x y Solutio: x +y x + x y+ xy + y 0 x +6x y xy +8y y Remembe that: the biomial expasio takes the fom: x + y x + x y + x y x y eplace the fist tem x with you fist tem; eplace the secod tem y with you secod tem emembe egatives; the coefficiets of x ad y ae also aised to a powe as well y y - is usually epeseted by just a egative sig. y Biomial theoem execise Go olie Q6: Use the biomial theoem to expad x + y Q66: Use the biomial theoem to expad x y Q6: Use the biomial theoem to expad x y

32 8 TOPIC. BINOMIAL THEOREM Q68: Expad usig the biomial theoem x + y Q69: Expad usig the biomial theoem x y Q0: Expad usig the biomial theoem x y Q: Expad usig the biomial theoem x y Ratioal coefficiets ae sometimes foud as is show i the followig activity. Biomial expasio: Ratioal coefficiets Example Poblem: Use the biomial theoem to expad x y Solutio: x x x x y x y + y+ + y + y 0 x 6 Remembe that: x y + x y xy + y the biomial expasio takes the fom: x + y x + x y + x y x y eplace the fist tem x with you fist tem; eplace the secod tem y with you secod tem emembe egatives; the coefficiets of x ad y ae also aised to a powe as well x x y Biomial theoem execise Go olie Q: Use the biomial theoem to expad x + y Q: Use the biomial theoem to expad x y

33 TOPIC. BINOMIAL THEOREM 9 Q: Use the biomial theoem to expad x + y.6 Fidig coefficiets Sometimes oly oe powe i a expasio is equied. By usig the geeal tem fomula give ealie, the coefficiet of ay tem i a expasio ca be foud. Examples. Poblem: Fid the coefficiet of x y i the expasio of x + y Solutio: Stat with the geeal tem of x + y which is: x y I this poblem,, so the geeal tem ow has the fom: x y But we wat: x y We eed to fid so we equate the powes: x x - So The This gives the tem: x y Replace with ad evaluate the coefficiet:!!!. Poblem: Fid the coefficiet of xy i the expasio of x y Solutio: The geeal tem i this poblem is: x y x y

34 0 TOPIC. BINOMIAL THEOREM The coefficiet is the Fo, we equie Theefoe, the coefficiet is:. Poblem: Fid the coefficiet of x i the expasio of x + x Solutio: The geeal tem i this poblem is: x x x x x We eed to use the ules of idices to simplify the powe of x. The coefficiet is the Fo, we equie. Theefoe, the coefficiet is: 8. Poblem: Fid the coefficiet of x 6 i the expasio of + x 8 Solutio: 8 The geeal tem is give by: 8 x 8 8 The coefficiet is the Fo 6, we equie x

35 TOPIC. BINOMIAL THEOREM Theefoe, the coefficiet is: 8 8!!! 6 Fidig coefficiets pactice Go olie Q: Fid the coefficiet of x i the expasio of x x Q6: Fid the coefficiet of x i the expasio + x Q: Fid the coefficiet of x y 6 i the expasio of x y 8 Fidig coefficiets execise Go olie Q8: Fid the coefficiet of x y i the expasio of x + y 6 Q9: Fid the coefficiet of x 9 i the expasio of + x Q80: Fid the coefficiet of x i the expasio of x + x Q8: Fid the coefficiet of x y i the expasio of x y Q8: Fid the coefficiet of x y i the expasio of x y Q8: Fid the coefficiet of x 6 i the expasio of + x 8 Q8: Fid the coefficiet of y i the expasio of y y

36 TOPIC. BINOMIAL THEOREM. Sigma otatio ad biomial theoem applicatios If a 0,a,..., a ae eal umbes, the the sum a 0 + a a is sometimes witte i shothad fom as a 0 Hee, the symbol, called sigma, meas 'the sum of'. The expessio 0 at the bottom of the sigma sig meas 'statig fom 0'. The lette at the top of the sigma sig meas 'util '. The tem a is the sequece of tems to be added togethe. Sigma otatio of biomial expasio x + y x y fo, N 0 x + x y + 0 x y + x y y Notice i the: st tem, is eplaced by 0 d tem, is eplaced by d tem, is eplaced by, ad so o. Example Poblem: Usig the biomial theoem expad x + y Solutio: x + y x y 0 x + x y + x y + x y + xy + 0 x +x +0x y +0x y +xy + y y The biomial theoem ca be used to fid powes of a eal umbe z. The techique is to split z ito two pats x ad y whee x is the closest itege to z ad y is the emaiig pat. Fo example, if z 9 the it ca be split as z 0

37 TOPIC. BINOMIAL THEOREM Examples. Evaluatig a decimal to a powe usig biomial expasio Poblem: Usig the biomial theoem, fid 0 Solutio: By witig 0 i the fom x + y, we ca use the biomial theoem to expad this whee x ad y Poblem: Usig the biomial theoem fid 0 Solutio: Remembe that: this expasio is fo x + y, so if you have x y, you have to emembe that the egative sig is pat of you umbe; thee ae may ways i which you ca split the umbe - it is ecommeded to choose oe of these umbes to be a whole umbe. Why is this?

38 TOPIC. BINOMIAL THEOREM Biomial applicatios pactice Go olie Q8: Usig the biomial theoem expad ad evaluate 0. Q86: Usig the biomial theoem expad ad evaluate. Biomial applicatios execise Go olie Ty this execise ow. Do ot use a calculato! Q8: Usig the biomial theoem expad ad evaluate 0 Q88: Usig the biomial theoem expad ad evaluate 0 6

39 TOPIC. BINOMIAL THEOREM.8 Leaig poits Biomial theoem Factoials!... fo N e.g.! 0!! e.g.!! 0! Biomial coefficiets The biomial coefficiet is!!! fo iteges, N ad <! e.g.!! 0 The biomial coefficiet ca be epeseted by C o Biomial coefficiets ae symmetical so the followig will give you the same aswe e.g e.g. + Pascal's tiagle Pascal's tiagle is made up of iteges set out as a tiagle. To fid the tems i the tiagle you follows these ules: The umbe oe appeas at the top ad the eds of each ow. To fid a umbe i the mai body, add the two umbes fom the above left ad above ight of it:

40 6 TOPIC. BINOMIAL THEOREM Whe the biomial coefficiets ae set up i a simila way ad evaluated, they fom Pascal's tiagle: Pascal's tiagle is a quick way to evaluate the biomial coefficiets. Biomial theoem The biomial theoem states that if x, y R ad N the: x + y x + x y + x y x y The geeal tem of x + y is give by: x y I both cases, the vaiables x ad y ca be eplaced by ay othe give tem. y Fidig coefficiets To fid the coefficiet of a give tem i the expessio x + y, we use the geeal tem x y equate the powes of x ad y i the give tem with that i the geeal tem to fid ; oce we have foud, we ca evaluate the biomial coefficiet ad ay othe coefficiets i the geeal tem. Sigma otatio ad biomial theoem applicatio To expad.0, we wite this umbe i the fom x + y, choosig x to be the itege closest to.0, the use the biomial expasio: x + y x + x y + x y x y y 0 eplacig x with, y with 0.0 ad with befoe evaluatig.

41 TOPIC. BINOMIAL THEOREM.9 Poofs Poof : Fist biomial coefficiet ule Pove that, the same as povig that!!!!! +!!!!!!! Poof : Secod biomial coefficiet ule + Pove that + I this poof, the followig facts ae equied:.!!. +! +!. Two factios ca be combied ove a commo deomiato. + +!!! +!!!!! +! + +!! +!! + +!! +!! + +! +!! +! +! +!! +! +

42 8 TOPIC. BINOMIAL THEOREM Poof : Biomial theoem Pove the biomial theoem which states that if x, y R ad, N, the: x + y x + x y + x y x y + 0 x y 0 This poof is by iductio, the method of which will be coveed late i the couse. y Let, the: LHS x + y x + y x + y RHS x y x + y x 0 y So LHS RHS ad the theoem holds fo Now, suppose that the esult is tue fo k whee k, the: k k k k k x + y k x k + x k y + x k y xy k + 0 k k Coside: x + y k+ x + yx + y k xx + y k + yx + y k k k k k k x k+ + x k y + x k y x y k + xy k + 0 k k k k k k k x k y + x k y + x k y xy k + 0 k k [ ] [ ] k k k k k x k+ + + x k y + + x k y [ ] [ ] k k k k k + x y k + + xy k + y k+ k k k k k k + k + x k+ + x k y xy k +y k+ k k+ k + x k+ y 0 y k y k+

43 TOPIC. BINOMIAL THEOREM 9 Hece, if it is tue fo k, it is also tue fo k +. Howeve, it was also tue fo so it is tue fo all values of. k k k + Remembe that fo all ad, ad that + 0 j + j j See the secod biomial coefficiet ule..0 Exteded ifomatio Pascal Blaise Pascal 6-66 was a Fech mathematicia, physicist ad philosophe. He published the tiagle i 'Taite du Tiagle Aithmetique' i 66 but did ot claim ecogitio fo it. His iteest i the tiagle aose fom his study of the theoy of pobabilities liked to his gamblig with his fied Femat. He was the iveto of the fist mechaical calculato ad a pogammig laguage pascal is amed afte him. Chu Shih - Chieh This Chiese mathematicia 0-0 published a vesio of the tiagle i his 'Pecious Mio of the Fou Elemets' i 0. Ope-eded challege Ty to ceate a -d vesio of Pascal's Tiagle usig a tiagula pyamid.

44 0 TOPIC. BINOMIAL THEOREM. Ed of topic test Ed of topic test Go olie Q89: a Simplify the followig sigma otatio: x b What ae the biomial coefficiets? c Wite dow the biomial expasio of x ad simplify you aswe. Q90: a Simplify the followig sigma otatio: 0 u v b What ae the biomial coefficiets? c Wite dow the biomial expasio of u v ad simplify you aswe. Q9: a Simplify the followig sigma otatio: 0 y b What ae the biomial coefficiets? c Wite dow the biomial expasio of y + ad simplify you aswe. Q9: a Simplify the followig sigma otatio: 0 k k b What ae the biomial coefficiets? c Wite dow the biomial expasio of k k ad simplify you aswe. Q9: 0 a Wite dow ad simplify the geeal tem i the expasio of x x 9. b Hece, o othewise, obtai the tem idepedet of x; that is, fid the tem that is a costat whee the powe of x is zeo.

45 TOPIC. BINOMIAL THEOREM Q9: a Wite dow ad simplify the geeal tem i the expasio of x x b Hece, o othewise, obtai the tem x. Q9: a Wite dow the biomial expasio of + x b Hece, calculate 0 9. Q96: a Wite dow the biomial expasio of + x b Wite dow the expessio fo i tems of + x ad evaluate. 0

46 GLOSSARY Glossay biomial coefficiet fomula The Biomial coefficiet fomula is!!! biomial coefficiet ule The fist biomial coefficiet ule is biomial coefficiet ule + The secod biomial coefficiet ule is + biomial theoem The Biomial Theoem states that if x, y R ad N the x + y x + x y + x y + + x y + + y geeal tem of x + y The geeal tem of x + y is give by x y factoial! called factoial is the poduct of the iteges,,,...,,, i.e.!... fo N

47 ANSWERS: UNIT TOPIC Aswes to questios ad activities Topic : Biomial theoem Revisio pactice page Q: y y + y y +y + y y +0y +9 y y +0y +9 y +0y +9 0y +60y +8y y +0y +9 0y +y y 9 Q: y y y y y 9y y +6 y 9y y +6 9y y +6 y y +8y 6y 96y +6 y 08y + y 6 Q: Hits: The sum of the poducts of the ie ad oute tems gives g + g g Aswe: g +g + g +g + Q: Hits: The sum of the poducts of the ie ad oute tems gives h +h h Aswe: h + h h h + Q: Hits: The simple commo facto is givig j + j 6 ad j + j 6 j +j Aswe: j +j j +j Q6: Hits: The simple commo facto is givig k k ad k k k +k Remembe that the sum of the poduct of the ie ad oute tems gives k + k k Aswe: k k k +k

48 ANSWERS: UNIT TOPIC Q: Fid the lowest commo multiple of +, which is : Q8: Fid the lowest commo multiple of 8, which is : Q9: 8 0 Q0: Q: Theefoe, Q: Theefoe, 0 8 Aswes fom page 8. Q: 6 Q:

49 ANSWERS: UNIT TOPIC Factoials calculato activity page 9 Expected aswe It is depedet upo the calculato that you have that will detemie which factoial you ca put ito you calculato without a eo beig displayed. Some calculatos ca oly give up to 69! so whe 0! is eteed, a eo is displayed. Howeve, we have to coside the accuacy of these aswes.! whe calculated by had gives , which is the same o the calculato. 6! by had gives , but o the calculato gives Thee is a vey subtle diffeece i the fouth last digit, but sice the scee of the calculato is limited, the umbes ae statig to be ouded. This illustates the eed to be awae of the limitatios of a calculato. Factoials pactice page 0 Q: 8! Q6:! Q: 8! ! 6! Q8: 0! ! 90 8! Q9:! 0 9! 0 9! Q0: 6! 6! 0!

50 6 ANSWERS: UNIT TOPIC Biomial coefficiets: C pactice page Q: Q: 6 Q: Q: C!!!!! 0 6!!! 6!! 6!!!!! 0!!! 6!! 6

51 ANSWERS: UNIT TOPIC Fist biomial coefficiet ule pactice page Q: Q6: Secod biomial coefficiet ule pactice page 6 Q: Q8:

52 8 ANSWERS: UNIT TOPIC Fidig give the value of a biomial coefficiet ad pactice page 6 Q9: 0!!! o Sice N the Q0:!!! o 6 Sice N the Q: 6!!! o 8 Sice N the 9

53 ANSWERS: UNIT TOPIC 9 Biomial coefficiets pactice page Q: Q: 00 Q: 00 99! Q: 0! Q6: 6 C Q: 9 Q8: Q9: Q0: Q: 6 6!!! 6!! 6 9!!! 9 8 6!! !!! 6!! 6 6 8

54 0 ANSWERS: UNIT TOPIC Q: 0 Biomial coefficiets execise page Q: 0 Q: Thee ae seveal aswes, sice the factoial is ot specified:! 6! 6!! 6! 0! 6! 80! 6! 0! 6! 00! Q: 8 Q6: Q: 8!!! 8 6!! Pascal's tiagle page 8 Q8: Row 0 Row Row Row Row 6 Row 0 0 Row Row

55 ANSWERS: UNIT TOPIC Q9: Row 0 Row Row Row Q0: Row Biomial coefficiets 0 Row 0 Row Row 0 6 Row Row Row 6 Row Values of biomial coefficiets Q: Whe the biomial coefficiets ae evaluated, they poduce Pascal's tiagle. Q: Q: 0 Q: 6

56 ANSWERS: UNIT TOPIC Q: Pascal's tiagle pactice page Q6: Usig the ule: Q: Usig the ule: Q8: Usig the ule: so Q9: Usig the ule: so Biomial theoem execise page 6 Q60: x + y 6 x 6 + x y + x y + x y + x y + xy x 6 +6x y +x y +0x y +x y +6xy + y 6 y 6

57 ANSWERS: UNIT TOPIC Q6: x + y 9 x 9 + x 8 y + x y + x 6 y + x y + x y + x y x y + xy 8 + y x 9 +9x 8 y +6x y +8x 6 y + 6x y + 6x y +8x y 6 +6x y +9xy 8 + y 9 Q6: x + y x + x y + x y + x y + xy + 0 x +x y +0x y +0x y +xy + y Q6: x + y x + x 6 y + x y + x y + x y + x y + xy x +x 6 y +x y +x y +x y +x y +xy 6 + y Q6: x + y 8 x 8 + x y + x 6 y + x y + x y + x y + x y xy + y 8 8 x 8 +8x y +8x 6 y +6x y +0x y +6x y +8x y 6 +8xy + y 8 y y Biomial theoem execise page Q6: x + y 0 x + x y+ x y + xy + y x + x y+6 x y + xy +y 8x +. x y+6.9x y +. xy + y 8x 08x y +x y xy + y Q66: x y x + x y+ x y + x y + 0 x y + y x + x y + 0 x y + 0 x y + x y + y x +x y+0 x y +0. x y + xy + y x x y 0x y 0.x y xy y

58 ANSWERS: UNIT TOPIC Q6: x y x + x y+ 0 x +x y+ y x xy +y y Q68: x +y x + x y+ xy + 0 x + x y+ x y + y 8x +6x y xy +y y Q69: x y x + x y+ x y + x y + 0 x + x y + 6 x y + x y + y 6x x y +x y 8xy + y y Q0: x y x + 0 y x y+ x y + x y + x 6 x 8 y+6x 9y x y +8y x 6 x y + x y xy +8y Q: x x x + x + + y 0 y y x 8 x y +x 9 y y x 8 9x y + x y y y

59 ANSWERS: UNIT TOPIC Biomial theoem execise page 8 Q: x + y x x x + y+ 0 x x + y+ y +y x 6 x + 6 x y ++ 9 xy + y Q: x y x + 0 x +x y + x xy + 6 y Q: x + y x + 0 x +x y x y + y y x y + +6x y y y +x + x + x y + x y + xy + 8 y x y + y + y y x + y Fidig coefficiets pactice page Q: The geeal tem i this poblem is: x x x x x We eed to use the ules of idices to simplify the powe of x. The coefficiet is the Fo, we equie The coefficiet is:

60 6 ANSWERS: UNIT TOPIC Q6: The geeal tem i this poblem is: The coefficiet is the Fo, we equie The coefficiet is: x x Q: 8 The geeal tem i this poblem is: 8 The coefficiet is the 8 x 8 y 8 8 x 8 y Fo 8, we equie 6 The coefficiet is: Fidig coefficiets execise page Q8: The geeal tem i this poblem is: 6 The coefficiet is the 6 x 6 y 6 x 6 y Fo 6, we equie The coefficiet is: Q9: The geeal tem i this poblem is: The coefficiet is the: x x

61 ANSWERS: UNIT TOPIC Fo 9, we equie The coefficiet is: 08 Q80: The geeal tem i this poblem is: x x x x x We eed to use the ules of idices to simplify the powe of x The coefficiet is the: Fo, we equie The coefficiet is: Q8: The geeal tem i this poblem is: The coefficiet is the: x y x y Fo, we equie The coefficiet is: Q8: The geeal tem i this poblem is: The coefficiet is the: x y x y Fo, we equie The coefficiet is: 6

62 8 ANSWERS: UNIT TOPIC Q8: The geeal tem i this poblem is: 8 The coefficiet is the: 8 8 x 8 x Fo 6, we equie The coefficiet is: 8 8!!! 6 Q8: The geeal tem i this poblem is: y y y y y We eed to use the ules of idices to simplify the powe of y. The coefficiet is the: Fo, we equie 0 The coefficiet is: 0 0 Biomial applicatios pactice page Q8:

63 ANSWERS: UNIT TOPIC 9 Q86: Biomial applicatios execise page Q8: Q88: Ed of topic test page 0 Q89: a x 0 x

64 60 ANSWERS: UNIT TOPIC b,,,,, which is equivalet to,, 6,, 0 c x + x + x + x + 0 6x + 8x + 6 x + x + 6x 60x + 600x 000x + 6 Q90: a u v u v 0 b,,,,,, which is equivalet to,, 0, 0,, 0 c u + u v+ u v + u v + 0 u v + v u + 6u v +0 u 9 v +0 u v + u 8v v u 9u v + 0u v 60uv + 0uv v Q9: a y + y 0 b,,,,,, which is equivalet to,, 0, 0,, 0 c y 0 + y 8 + y 6 + y + y + 0 y 0 + y 8 +0 y 6 +0 y + y + y 0 +y y 6 + 0y + 00y + 680

65 ANSWERS: UNIT TOPIC 6 Q9: a k k 0 k k 0 k b,,,,, which is equivalet to,, 6,, 0 c k + k + k + k + 0 k +k + 6 k k k 0k + 0 k 00 k + 6 k 8 Q9: a Steps:? What is the geeal tem fo x??? 9 x 9 x? x? What is the geeal tem fo? x???? 9 Aswe: 9 x 9 b Steps: Whe does the powe of x equal 0? Fo a tem to be idepedet of x, the 9 0, i.e. 9 Substitute ito 9 x 9 Aswe: 9 6 x k 8

66 6 ANSWERS: UNIT TOPIC Q9: a Steps:? What is the geeal tem fo?? What is the geeal tem fo? 0 Aswe: 0 x 0 b Steps: x??? x? x??? Whe does the powe of x equal? Fo the tem x, the 0, i.e. 0 Substitute ito 0 x 0 0 x 0 x Aswe: 0 8 x 66 x 80980x Q9: a Hits: Use Pascal's tiagle fo the coefficiets o the biomial expasio which is: x 0 Aswe: + x + x + x + x + x + 0 +x +0x +0x +x + x b Steps: What is the expessio fo 0 9 i tems of + x? + 0 x

67 ANSWERS: UNIT TOPIC 6 Aswe: Q96: a Hits: Use Pascal's tiagle fo the coefficiets o the biomial expasio which is: x 0 Aswe: + x + x + x + x + 0 +x +6x +x + x b x 0