The Binomial Theorem

Transcription

1 The Biomial Theorem Robert Marti Itroductio The Biomial Theorem is used to expad biomials, that is, brackets cosistig of two distict terms The formula for the Biomial Theorem is as follows: (a + b ( k k0 a + a k b k ( a b + ( ( a b + + ab + b Note that we could write ( 0 ad ( as the coefficiets for a ad b respectively, but these coefficiets are equal to oe so we have omitted them Oe will eed to kow this formula, or at least kow how to derive it o the spot Aother formula which is very useful ad should be remembered, is this (derived at the ed: ( + x + x + ( x + ( ( x 3 + 3! Basic ituitio ( ( ( 3 x 4 + 4! Cosider the expressio (a + b(c + d How might we multiply these brackets together? Hopefully, you should be able to expad this quite aturally: (a + b(c + d ac + ad + bc + bd What have we actually doe here? We have picked oe term from the first bracket, the multiplied it by a term we picked from the secod bracket Above, the ac term is made from pickig a from the first bracket ad c from the secod bracket, the multiplyig

2 We the repeat this for all possible ways of pickig terms, the add them up to give the fial expasio The result of this is that everythig is multipled by everythig ; the fial expressio cosists of terms each formed by the multiplicatio of exactly oe compoet from each bracket This ( bit of ituitio is very importat, ad will later explai why the biomial coefficiet, k (or C k if you prefer shows up 3 Some simple expasios 3 The expaisio of ( + x Let us begi with oe of the simplest possible biomials, ( + x Expadig ( + x like we did with (a+b(c+d, we get +x+x+x (which is the simplified to +x+x How do we explai this result with regards to pickig terms? Remember, we ca pick oe term from the first bracket ad oe term from the secod bracket (it is implicit that we the multiply I the fial expasio: The comes from pickig a from each bracket The x terms come from pickig a x from a bracket ad pickig a from the other The x term comes from pickig a x from both brackets There are two x terms because there are two possible brackets from which we could pick the x: we ca either pick the x from the first bracket, or from the secod bracket Geerally, there is more tha oe way we ca produce a certai power of x These eed to be accouted for 3 The expaisio of ( + x 3 If we write out the brackets fully, we have ( + x( + x( + x We kow that the expasio will have a costat, a x, a x, ad a x 3 term The trouble is that we do t kow the coefficiets (the costat umbers i frot of each power of x We will therefore say, for ow, that: (We just eed to fill i the blaks! ( + x 3 _ + _x + _x + _x 3 It is clear that the costat term is formed as a result of pickig from each of the brackets Sice, this coefficiet is clearly

3 How do we get the x term? Now thigs are gettig iterestig Clearly, from the three brackets, we eed to pick oe x ad two s, because x x The thig is, there are three possible brackets we ca choose our x from: but we oly eed oe! So how may ways ca we choose oe x ad two s? At this poit, you must rememer your combiatorics We have three, ad we eed to choose oe (order does t matter The aswer is clearly ( 3 What about the x term? Applyig the same reasoig as above, the x term is formed from multiplyig two x terms ad a : x x x How may ways, from the three brackets, ca we pick two x terms? The aswer is simply ( 3 The x 3 term obviously comes from multiplyig three x terms together, ie we eed to choose a x from each of the three brackets There is oly oe way we ca do this, so the last coefficiet is Puttig this all together ad computig the biomial coefficiets, we arrive at the expasio: ( ( 3 3 ( + x 3 + x + x + x 3 + 3x + 3x + x 3 Problem What is the coefficiet of x 4 i the expasio of ( + x 7? Questios askig for specific coefficiets are commo, ad ofte uderpi the solutios to more complex biomial questios It is thus critical that you kow how to do this type of questio Imagie writig out all the brackets ext to each other, like this: ( + x( + x( + x( + x( + x( + x( + x The x 4 term will be made by multiplyig four x terms together (ad obviously three remaiig s How may ways ca we choose four x terms, rememberig that there are a total of seve to choose from? Simple The biomial coefficiet will be ( A slight modificatio to the problem I the above problem, what if I had asked for the coefficiet of x 4 i the expasio of (+x 7? How would the problem chage? Very little, it turs out I am goig to rewrite the above explaatio, with some mior amedmets The x 4 term will be made by multiplyig four x terms together (ad obviously three remaiig s How may ways ca we choose four x terms, rememberig that there are a total of seve to choose from? Simple The biomial coefficiet will be ( 7 4 Remember that after pickig everythig, we have to multiply (this has bee implicit i everythig 3

4 we ve doe But let s explicitly do this ow We picked four x terms ad three terms, so we cosider x x x x, which is writte more compactly as ( 3 (x 4 But before we fiish, we must remember to multiply by the biomial coefficiet The fial x 4 term is therefore ( 7 4 ( 3 x 4, so the coefficiet is ( 7 4 ( 3 4 Applyig your kowledge Problem Fid the coefficiet of x i the expasio of (3x 8 Out of 8 brackets, we pick five 3x s ad three of the ( terms The biomial coefficiet is thus 8 choose, or ( 8 Combiig all this, we kow that the x term will be ( 3 (3x Ask your calculator for the aswer; it should say 0884 ( 8 Problem 3 Fid the coefficiet of a b 7 i the expasio of (a + b We ( eed to choose five a terms out of the twelve brackets, so the biomial coefficiet is ( Note that we could istead say 7 ; both of these coefficiets are equal (ca you thik why? Problem 4 Fid the term cotaiig x 0 i the expasio of ( + x 7 This looks asty, but actually is t To make x 0, we eed to choose five x terms (sice x multiplied by itself five times gives x 0 The biomial coefficiet is thus ( 7 Hece the term i questio will be ( 7 ( (x You ca quickly check for yourself that expadig this will give you the required x 0 term Problem Fully expad (a + b There will be seve terms i this expasio How do I kow? Well, the first term will be made from choosig six a s The ext term will be made from choosig five a s ad oe b The third term will be made from choosig four a s ad two b s Ad so o Hece, the expasio is: a + ( a b + ( a 4 b + 4 ( a 3 b ( a b 4 + ( ab + b I doubt that the IB is mea eough to ask you to do this just for the lols, but it may help you for the ext questio 4

5 Problem Determie the costat term i the expasio of (x + x Now is the time to use your brai! This is a typical problem that comes up i both SL ad HL papers Both terms i our biomial cotai x, so the costat term is certaily ot x We eed to fid some particular choice of terms that will allow the x s to cacel Easier said tha doe What happes if we choose two x terms ad four terms? Well, the biomial coefficiet x is clearly ( ( Thus, the term i the expasio will be (x ( 4 or ( x x Simplifyig x 8 usig the laws of expoets, we get We wated x 0, but istead got x This meas x that we chose too may of the terms x What about havig three x terms ad three terms? The biomial coefficiet will just x be ( ( 3 The associated term i the expasio will be 3 (x 3 ( 3, which simplifies to x 0 We wated x 0, but istead got x 3 Too egative still x 3 We will ow make a more sesible choice: four x terms ad two terms The biomial x coefficiet is ( ( 4 Thus, the term i the expasio will be 4 (x 4 ( or ( x 4 x 4 Somethig special happes here The x s all cacel out, leavig a loe ( x 4 4 The costat term is thus To summarise, fidig the costat term is a bit of a balacig act, you eed to choose just the right umber of x a terms such that they cacel out with the terms This x b ivolves a small bit of guesswork, which ca be doe metally quite fast Problem 7 What is the costat term i the expasio of (3x x 9? Practice makes perfect let s try this agai We immediately kow that we are goig to eed more of the x terms, because each 3x term has two x s i it At this poit, you have to thik of the possiblities If I take two 3x terms ad seve ( x terms, the fial result will have x Too low If I take three 3x terms ad six ( x terms, oh look, the result will have x 0 Hece the term we eed is ( 9 3 (3x 3 ( x As we discussed, the x s will cacel out The umbers, however, will still remai Thus the coefficiet will be ( ( Problem 8 Fid the x term i the expasio of ( + x 4 ( + x 7 Please do t expad everythig, uless you wat to amuse a HL maths studet stadig behid you What you eed to do is fid the first few terms (up to x i each of the

6 brackets I trust you ll be able to do this by ow ( + x 4 ( + x 7 ( + 3x + 4x + ( + 7x + x + Now, we thik We eed a x how ca we make this? Remember, whe multiplyig brackets, everythig is multiplied by everythig So, somewhere i the expasio, the from the first bracket will be multiplied by the, or the 7x, or the x (ad so o The questio is, which of these will yield us a x? Obviously, it will be x We ow look at the ext term i the first bracket, the 3x Agai, this will be multiplied by every term i the secod bracket but which oe gives us a x? Obviously, 3x 7x Proceedig, we examie the 4x With which term i the secod bracket should we multiply this to result i a x? Well, we multiply it by the costat term sice 4x already cotais a x Thus, 4x To summarise: x 33x 3x 7x 4x 4x 4x That is to say, whe we multiply out ( + x 4 ( + x 7, the expasio will cotai the above terms (ad may others with differet powers of x ( + x 4 ( + x x + 4x + 4x + But do t just stad there, simplify! All of these x terms ca be combied to give 84x Hece, the coefficiet of x is 84 Quite easily doe Here is the fial compiled solutio, which is what I d write i a exam: ( + x 4 ( + x 7 ( + 3x + 4x + ( + 7x + x + x 33x 3x 7x 4x 4x 4x the coefficiet of x is Extesio Problem Fid the first four terms i the expasio of + x Ulikely to come up i SL, but you ever kow We start by fidig a geeral formula for the expasio of ( + x, the later substitute i

7 The Biomial Theorem states that: ( (a + b a + a b + ( a b + ( a 3 b b 3 (We eed oly cosider the first four terms Sice we wat the expasio of ( + x, we substitute a ad b x Thus: ( ( ( ( + x + x + x + x x 3 As promised, we ow just substitute i But if you try fidig ( 0, your calculator will retur a error message Fair eough: how ca we choose oe thig from a selectio of 0 thigs the questio makes o sese! We will delay substitutig, ad istead try to fid icer expressios for the combiatorial coefficiets ( (, (, ad 3 Recall the actual formula for ( r : ( r! r!( r! Notice that this ca be writte differetly, by expadig the factorials! ( ( ( r + ( r( r r!( r! r!( r( r ( r Both the top ad bottom of the fractio cotai ( r( r terms, which ca be cacelled Thus: ( ( ( ( r + r r! If we substitute r,, 3 respectively: (! (! (! (!(! (! ( ( 3 3!( 3! This derives the very useful formula for the geeral expasio of ( + x, which was stated i the itroductio, ad works for o-iteger values of Now, ad oly ow, may we substitute ( + x / + x + /( / + x 8 x + x3 + x + /( /( 3/ x + 3! This is actually quite a remarkable result The square root of + x ca be expressed as a ever-edig polyomial Is t life great? 7