P1 Chapter 8 :: Binomial Expansion

Transcription

1 P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk Last modified: 6 th August 7

2 Use of DrFrostMaths for practice Register for free at: Practise questios by chapter, icludig past paper Edexcel questios ad extesio questios (e.g. MAT). Teachers: you ca create studet accouts (or studets ca register themselves).

3 Chapter Overview :: Pascal s Triagle :: Factorial Notatio Give that value of a. 8 = 8!, fid the!a! :: Usig expasios for estimatio :: Biomial Expasio Use your expasio to estimate the Fid the first terms i the value of.5 to 5 decimal places. biomial expasio of + 5x, givig terms i ascedig powers of x.

4 Starter a) Expad a + b b) Expad a + b c) Expad a + b d) Expad a + b e) Expad a + b a + b a + ab + b a + a b + ab + b a + a b + 6a b + ab + b What do you otice about: The coefficiets: The powers of a ad b: They follow Pascal s triagle (we ll explore o ext slide). Power of a decreases each time (startig at the power) Power of b icreases each time (startig at )

5 More o Pascal s Triagle The secod umber of each row tells us what row we should use for a expasio. So if we were expadig + x, the power is, so we use this row. 6 5 I Pascal s Triagle, each term (except for the s) is the sum of the two terms above. Fro Tip: I highly recommed memorisig each row up to what you see here. We ll see later WHY each row gives us the coefficiets i a expasio of a + b

6 Example Fid the expasio of + x First fill i the correct row of Pascal s triagle. + x = + Simplify each term (esurig ay umber i a bracket is raised to the appropriate power) ( ) x x x x Next have descedig or ascedig powers of oe of the terms, goig betwee ad (ote that if the power is, the term is, so we eed ot write it). Ad do the same with the secod term but with powers goig the opposite way, otig agai that the power of term does ot appear. = x + 6x + 6x + 8x Fro Tip: Iitially write oe lie per term for your expasio (before you simplify at the ed), as we have doe above. There will be less faffig tryig to esure you have eough space for each term.

7 Aother Example ( x) is the same as ( + x ), so we expad as before, but use x for the secod term. x = ( ) x x x = 6x + x 8x Fro Tip: If oe of the terms i the origial bracket is egative, the terms i your expasio will oscillate betwee positive ad egative. If they do t (e.g. two cosecutive egatives), you re doe somethig wrog!

8 Gettig a sigle term i the expasio The coefficiet of x i the expasio of cx 5 is 7. Fid the possible value(s) of the costat c. The 5 row i Pascal s triagle is 5 5. If we cout the as the th term, we wat the d term, which is. Sice we wat the x term: The power of cx must be. The power of must be (the two powers must add up to 5). Therefore term is: cx = 8c x 8c = 7 c = 9 c = ±

9 Test Your Uderstadig Edexcel C

10 Exercise 8A Pearso Pure Mathematics Year /AS Pages 6-6 Extesio [MAT 9 J] The umber of pairs of positive itegers x, y which solve the equatio: x + 6x y + xy + 8y = is: A) B) 6 C) 9 D) + The LHS is the biomial expasio of x + y, therefore: x + y = x + y = x = 9 y I order for x to be a positive iteger, y ca be betwee ad 9. The aswer is C.

11 Factorial ad Choose Fuctio!! = said factorial, is the umber of ways of arragig objects i a lie. For example, suppose you had three letters, A, B ad C, ad wated to arrage them i a lie to form a word, e.g. ACB or BAC. There are choices for the first letter. There are the choices left for the secod letter. There is the oly choice left for the last letter. There are therefore =! = 6 possible combiatios. Your calculator ca calculate a factorial usig the x! butto.! Cr = r =! r! r! said choose r, is the umber of ways of choosig r thigs from, such that the order i our selectio does ot matter. These are also kow as biomial coefficiets. For example, if you a football team captai ad eed to choose people from amogst i your class, there are =! = possible selectios.!6! (Note: the otatio is preferable to C ) Use the Cr butto o your calculator (your calculator iput should display C )

12 Examples Calculate the value of the followig. You may use the factorial butto, but ot the Cr butto. a) 5! 5 b) c)! d) e) f) g) 8 a b c d e f g 5! = 5 = 5 = 5!!! = 6 =! =. Accept this for the momet, but all will be explaied i part (e). Coceptually, there is clearly ways to choose thig from. But usig the formula: =! =!9!! = for all. We d expect there to be way to choose o thigs (sice o selectio is itself a possibility we should cout). Usig the formula: =! =!! This provides justificatio for lettig! =.! = =! = = = 9! 8!! 8 7!! = for all. 8 =!. This is the same as above. I geeral, where the 8!! bottom umber is above half of the top, we ca subtract it from the top, i.e. =.

13 Why do we care? If the power i the biomial expasio is large, e.g. x +, it is o loger practical to go this far dow Pascal s triagle. We ca istead use the choose fuctio to get umbers from aywhere withi the triagle. We ll practise doig this after the ext exercise. Notice: The top umber matches the row umber. The bottom umber goes from ad evetually matches the top umber. It s easy to see from the symmetry of Pascal s Triagle that = for example. 6 th row st row d row rd row Textbook Note: The textbook refers to the top row as the st row ad the first umber i each row as the st etry. This might soud sesible, but is agaist accepted practice: It makes much more sese that the row umber matches the umber at the top of the biomial coefficiet, ad the etry umber matches the bottom umber. We therefore call the top row the th row ad the first etry of each row the th etry. So the kth etry of the th row of Pascal s Triagle is therefore a ice clea k, ot as suggested by the k textbook.

14 Extra Cool Stuff (You are ot required to kow this, but it is helpful for STEP) We earlier saw that each etry of Pascal s Triagle is the sum of the two above it. Thus for example: + = More geerally: k + = k k This is kow as Pascal s Rule. Iformal proof of Pascal s Rule: Suppose I have items ad I have to choose k of them. Clearly there s possible selectios. k But we could also fid the umber of selectios by cosiderig the first item of the available: It might be chose. If so, we have k items left to choose from amogst the remaiig. That s possible selectios. k Otherwise it is ot chose. We still have k items to choose, from amogst the remaiig items. That s possible selectios. k Thus i total there are k + possible selectios. k

15 Exercise 8B Pearso Pure Mathematics Year /AS Pages 6

16 Usig Biomial Coefficiets to Expad I the previous sectio we leart about the choose fuctio ad how this related to Pascal s Triagle Why do rows of Pascal s Triagle give us the coefficiets i a Biomial Expasio? Oe possible selectio of terms from each bracket. Cosider: a + b 5 = a + b a + b a + b a + b (a + b) Each term of the expasio ivolves pickig oe term from each bracket. How may times will a b appear i the expasio? To get a b we must have chose a s from the 5 brackets (the rest b s). 5 That s ways, givig us 5 a b i the expasio of a + b 5.

17 Usig Biomial Coefficiets to Expad! The biomial expasio, whe N: N is the set of atural umbers, i.e. positive itegers. This formula is oly valid for positive itegers. I Year you will see how to deal with fractioal/egative. a + b = a + a b + a b + + r a r b r + + b Fid the first terms i the expasio of x +, i ascedig powers of x. x + = ( ) x x x + This is exactly the same method as before, except we ve just had to calculate the Biomial coefficiets ourselves rather tha read them off Pascal s Triagle. = + x + 5x + x +

18 Test Your Uderstadig Fid the first terms i the expasio of x 7, i ascedig powers of x. x 7 = Fro Note: The + idicates that there would have bee other terms i the expasio = 8 8 x + x + +

19 Exercise 8C Pearso Pure Mathematics Year /AS Page 6 Extesio [AEA Qa] I the biomial expasio of + the coefficiets of x ad x are equal ad o-zero. Fid the possible values of. 5 x Hit: Remember that =! Ca you similarly simplify usig r =!? r! r! [STEP I Q5a] By cosiderig the expasio of + x, where is a positive iteger, or otherwise, show that: = + x = + x + x + + x Lettig x = gives the desired result. Froflectio: This meas that the sum of each row i Pascal s Triagle gives successive powers of. Safe!

20 Gettig a sigle term i the expasio I the expasio of a + b the geeral term is give by r a r b Expressio Power of x i term wated. Term i expasio a + x x 75 5 x 7 x + 6 a7 x Note: The two x x 6 x powers add up to.

21 Gettig a sigle term i the expasio The coefficiet of x i the expasio of + qx is 6. Fid the possible value(s) of the costat q. Term is: 6 qx = q x Therefore: q = 6 q = 6 q = ±

22 Test Your Uderstadig I the expasio of + ax, where a is a o-zero costat the coefficiet of x is double the coefficiet of x. Fid the value of a. x term: x term: 8 ax = 5a x 7 ax = a x But a is o-zero, so a = a = 5a a = 9a a a = a a = a = or a =

23 Exercise 8D Pearso Pure Mathematics Year /AS Pages Extesio [MAT G] Let be a positive iteger. The coefficiet of x y 5 i the expasio of + xy + y equals: A) B) C) D) E) 8 5 Try to imagie brackets writte out. To get x y 5, we must have chose xy from brackets, y from oe ad from the remaiig brackets. That s choices for the xy term, ad choices for the y term. Usig the defiitio of the choose fuctio, you ca show that = [STEP I Q6] By cosiderig the coefficiet of x r i the series for + x + x, or otherwise, obtai the followig relatio betwee biomial coefficiets: r + r = + r Notig that + x + x = + x +, the x r + term is x r. r But the x r term could be obtaied either by i the first bracket multiplied by x r term i the secod, givig r xr, or the x i the first bracket multiplied by the x r term i the secod, x r xr = r xr. Thus comparig coefficiets: r + + = r r

24 Estimatig Powers Edexcel C Ja Q Fro Tip: Use your calculator to compare agaist the exact value of.5 8. a + x 8 = x x x 5 = + x + 7 x x + b Comparig. 5 8 to + x 8, the:. 5 = + x. 5 = x x =. Usig our expasio with x =. : =. 8 to dp Why should this be a reasoably good approximatio of.5 8 despite the missig terms i the expasio? x r becomes icreasigly small whe x < as the power icreases. Thus the. terms ad beyod will be egligibly small.

25 Test Your Uderstadig Edexcel C Ja 8 Q

26 Exercise 8E Pearso Pure Mathematics Year /AS Page 68-69