Math 105 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII. 7. Binomial formula. Three lectures ago ( in Review of Lectuires IV ), we have covered

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1 Math 5 TOPICS IN MATHEMATICS REVIEW OF LECTURES VII Istructor: Lie #: 59 Yasuyuki Kachi 7 Biomial formula February 4 Wed) 5 Three lectures ago i Review of Lectuires IV ) we have covered / \ / \ / \ / \ / \ / \ 3 3 / \ / \ / \ / \ / \ / \ / \ / \ / \ 5 5 / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ row ) row ) row ) row 3) row 4) row 5) row 6) row 7) This is called the Pascal s triagle How the umbers are arraged i the Pascal s triagle is dictated by the rule:

2 Rule At every spot that umber equals the sum of two umbers right above it Meawhile let s recall two formulas ) x y x x y y ) 3 x y x 3 3 x y 3 x y y 3 The first oe from Review of Lectures V the secod oe from Review of Lectures VI ) Today I wat to bridge these two subjects As a starter let s address what comes ext to squarig ad cubig Higher powers We have defied x ad x 3 : x x x ad x 3 x x x ) ) the square of x the cube of x It is very atural to exted this ad cosider x x x x x x x 3 x x x x 4 x x x x x 5 x x x x x x 6 x x x x x x x 7 x x x x x x x

3 So here is the formal defiitio: Defiitio For a positive iteger defie x as x x x x x This is proouced as x to the -th power x raised to the power of or simply x to the Example For 4 x 4 x x x x x x x x x x x x x x x x x x x x x x x x x to the 4 ) Let me repeat that for ad we set x ad x x ad these are by covetio 3

4 Example Example Example Example Example Example Example Example Example Example Example 4 te thousad ) 5 oe hudred thousad ) 6 oe millio ) 7 te millio ) 8 oe hudred millio ) 9 ) oe billio More geerally for a positive iteger I mathematics we do t place comma ) after every third digit 4

5 -to-the-powers We have Well othig impressive or dramatic -to-the-powers We have Agai othig impressive or dramatic By the way whereas is udefied 5

6 ) -to-the-powers I short ) We have ) ) ) 3 ) 4 ) 5 ) 6 ) 7 ) 8 ) 9 ) if is eve if is odd ) ) Exercise Fid each of ) 5 ) 48 9 ) [ Aswers ] : ) 5 ) 48 9 ) 6

7 ) a -to-the-powers I short a) We have a) a a) a a) 3 a 3 a) 4 a 4 a) 5 a 5 a) 6 a 6 a) 7 a 7 a) 8 a 8 a) 9 a 9 a) a a if is eve a if is odd ) ) Exercise Fid each of ) 6 ) ) [ Aswers ] : ) 6 ) ) 65 7

8 -to-the-powers The umbers i the followig sequece are called -to-the-powers : to-the-powers frequetly appear i mathematics Please familiarize yourself with the above listed umbers the first sixtee of -to-the-powers ) Exercise 3 Idetify all -to-the-powers amog the umbers listed below Write each of those -to-the-powers i the form with a cocrete positive iteger

9 [ Aswer ] : Biomial Formula Fially today s mai theme Check this out: ) ) 3) 4) 5) 6) x y) x y x y) x x y y x y) 3 x 3 3x y 3 x y y 3 x y) 4 x 4 4x 3 y 6 x y 4 x y 3 y 4 x y) 5 x 5 5x 4 y x 3 y x y 3 5 x y 4 y 5 x y) 6 x 6 6x 5 y 5x 4 y x 3 y 3 5x y 4 6xy 5 y 6 The first oe is just a tautology The ext two we have covered i our last two lectures Others look ew to you Let s dissect First we have a coveiet word for the uderlied umbers i the above They are called the coefficiets Also for example x 3 ca be regarded as a abbreviatio for x 3 etc So we say that i the right-had side of each of the above lies the first ad the last terms both have coefficiet Thus: The coefficiet of x 3 y i the right-had side of 4) is 4 The coefficiet of x y 3 i the right-had side of 5) is The coefficiet of y 6 i the right-had side of 6) is 9

10 Now those uderlied umbers coefficiets ) look familiar If we just pick up those coefficiets from left to right i each lie: ) ) 3) 3 3 4) ) 5 5 6) Yes ideed: This is exactly the Pascal s triagle So ca you guess the formula for 7 xy) 8 xy) ad each? Yes accordig to Pascal the list of coefficiets cotiues as 7) ) Accordigly 7) ) 7 x y x 7 7x 6 y x 5 y 35x 4 y 3 35x 3 y 4 x y 5 7xy 6 y 7 8) ) 8 x y x 8 8x 7 y 8x 6 y 56x 5 y 3 7x 4 y 4 56x 3 y 5 35x y 6 8xy 7 y 8 These are ideed the correct formulas Now at this poit I m sure you already kow how to form the correct formula for xy) for 9 I wat to officially formulate it That requires me to itroduce oe ew otatio

11 Notatio biomial coefficiet) I the Pascal s triagle page ) : The umbers i row from left to right are deoted as ) ) The umbers i row from left to right are deoted as ) ) ) The umbers i row 3 from left to right are deoted as ) ) ) ) The umbers i row 4 from left to right are deoted as ) ) ) ) ) The umbers i row 5 from left to right are deoted as ) ) ) ) ) )

12 More geerally i the Pascal: The umbers i row from left to right are deoted as ) ) ) ) 3 ) These are called the biomial coefficiets Note ) ) Example 3 ) 7 ) 6 ) 7 5 ) 6 6 ) ) 6 3 ) As for the formula for ext page ) k for geeral ad k see Formula A i the Usig this ew otatio we ca rewrite the previous formulas as ) ) ) x y x y ) ) ) ) ) x y x ) xy y ) ) ) ) ) x y x ) 3 x y xy y 3 3 ) ) ) ) ) ) x y x ) 4 x 3 y x y xy 3 y Do you see the patters? More geerally the right formula for xy) is give below Formula B i the ext page )

13 Formula A biomial coefficiets) Let ad k be itegers with < k < The ) k ) ) ) k 3 k Formula B Biomial Formula) Let be a positive iteger The ) xy ) x ) x y ) x y ) x 3 y 3 3 ) x 3 y 3 3 ) x y ) xy ) y Exercise 4 Spell out each of the followig biomial coefficiets i the fractio form You do t have to calculate the aswers ) 9 5 ) 7 ) 6 ) 4 ) 5 ) 8 [ Aswers ] : ) ) ) ) ) 5 )

14 Exercise 5 Spell out the biomial formula for each of ) 5 ) 8 9 a) x y b) x y ad c) x y) ) Ieachofa)b)c) firstgivetheformulathaticludestheotatio ) k covert those ito umbers ad rewrite your aswer accordigly k [ Aswers ] : a) ) 5 x y The ) 5 x 5 ) 5 x 4 y ) 5 x 3 y ) 5 x y 3 3 ) 5 xy 4 4 ) 5 y 5 5 x 5 5 x 4 y x 3 y x y 3 5 xy 4 y 5 b) ) 8 x y ) 8 x 8 ) 8 x 3 y 5 5 ) 8 x 7 y ) 8 x y 6 6 ) 8 x 6 y ) 8 xy 7 7 ) 8 x 5 y 3 3 ) 8 y 8 8 ) 8 x 4 y 4 4 x 8 8 x 7 y 8 x 6 y 56 x 5 y 3 7x 4 y 4 56x 3 y 5 8 x y 6 8 xy 7 y 8 c) ) 9 x y ) 9 x 9 ) 9 x 4 y 5 5 ) 9 x 8 y ) 9 x 3 y 6 6 ) 9 x 7 y ) 9 x y 7 7 ) 9 x 6 y 3 3 ) 9 xy 8 8 ) 9 x 5 y 4 4 ) 9 y 9 9 x 9 9 x 8 y 36 x 7 y 84 x 6 y 3 6x 5 y 4 6x 4 y 5 84 x 3 y 6 36 x y 7 9 xy 8 y 9 4

15 Pop quiz How much does it make it you add up the umbers i oe whole row i Pascal? Let s experimet: Row : Row : 4 Row 3: Row 4: Row 5: Row 6: Row 7: To alig the aswers: So these umbers look familiar right? Yes they are -to-the-powers More precisely: Fact The sum of the umbers i the -th row of Pascal equals What I mea by this is that it is ot just for Row Row 7 but if you do the same for the lower rows the same is always true But why? The clue is this is a simple applicatio of the Biomial Formula Formula B above ) I will leave it as your ow exercise to figure it out If you eed further clue: Substitutig some appropriate umber for each of x ad y i the Biomial Formula would do it So figure out those umbers to be substituted for each of x ad y 5

16 Exercise 6 Explai why the followig fact follows from the Biomial Theorem: Fact The sum of the umbers i the -th row of Pascal equals Idicate what umber to substitute for each of x ad y i the Biomial Formula [ Aswer ] : Substitutig x ad y i the Biomial Formula yields ) ) ) ) ) ) ) ) ) ) ) ) ) I short ) ) ) ) ) ) 6