652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you will study higher powers of iomials. Some Examples We kow that (x y) 2 x 2 2xy y 2. To fid (x y) 3, we multiply (x y) 2 y x y: (x y) 3 (x 2 2xy y 2 )(x y) (x 2 2xy y 2 )x (x 2 2xy y 2 )y x 3 2x 2 y xy 2 x 2 y 2xy 2 y 3 x 3 3x 2 y 3xy 2 y 3 The sum x 3 3x 2 y 3xy 2 y 3 is called the iomial expasio of (x y) 3. If we agai multiply y x y, we will get the iomial expasio of (x y) 4. This method is rather tedious. However, if we examie these expasios, we ca fid a patter ad lear how to fid iomial expasios without multiplyig. Cosider the followig iomial expasios: (x y) 0 1 (x y) 1 x y (x y) 2 x 2 2xy y 2 (x y) 3 x 3 3x 2 y 3xy 2 y 3 (x y) 4 x 4 4x 3 y 6x 2 y 2 4xy 3 y 4 (x y) 5 x 5 5x 4 y 10x 3 y 2 10x 2 y 3 5xy 4 y 5 Oserve that the expoets o the variale x are decreasig, whereas the expoets o the variale y are icreasig, as we read from left to right. Also otice that the sum of the expoets i each term is the same for that etire lie. For istace, i the fourth expasio the terms x 4, x 3 y, x 2 y 2, xy 3, ad y 4 all have expoets with a sum of 4. If we cotiue the patter, the expasio of (x y) 6 will have seve terms cotaiig x 6, x 5 y, x 4 y 2, x 3 y 3, x 2 y 4, xy 5, ad y 6. Now we must fid the patter for the coefficiets of these terms. Otaiig the Coefficiets If we write out oly the coefficiets of the expasios that we already have, we ca easily see a patter. This triagular array of coefficiets for the iomial expasios is called Pascal s triagle. 1 (x y) 0 1 1 1 (x y) 1 1x 1y 1 2 1 (x y) 2 1x 2 2xy 1y 2 1 3 3 1 (x y) 3 1x 3 3x 2 y 3xy 2 1y 3 1 4 6 4 1 (x y) 4 1x 4 4x 3 y 6x 2 y 2 4xy 3 1y 4 1 5 10 10 5 1 Coefficiets i (x y) 5 Notice that each lie starts ad eds with a 1 ad that each etry of a lie is the sum of the two etries aove it i the previous lie. For istace, 4 3 1,
12.5 Biomial Expasios (12-27) 653 ad 10 6 4. Followig this patter, the sixth ad seveth lies of coefficiets are 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 Pascal s triagle gives us a easy way to get the coefficiets for the iomial expasio with small powers, ut it is impractical for larger powers. For larger powers we use a formula ivolvig factorial otatio.! ( factorial) If is a positive iteger,! (read factorial ) is defied to e the product of all of the positive itegers from 1 through. calculator close-up You ca evaluate the coefficiets usig either the factorial otatio or Cr. The factorial symol ad Cr are foud i the MATH meu uder PRB. For example, 3! 3 2 1 6, ad 5 4 3 2 1 120. We also defie 0! to e 1. Before we state a geeral formula, cosider how the coefficiets for (x y) 4 are foud y usig factorials: 4! 4 3 2 1 1 Coefficiet of x 4 (or x 4 y 0 ) 4!0! 4 3 2 1 1! 4 3 2 3!1! 3 2 1 1 1 Coefficiet of 4x 3 y! 4 3 2 1 6 2!2! 2 1 2 1 Coefficiet of 6x 2 y 2! 4 3 2 1!3! 1 3 1 2 1 Coefficiet of 4xy3! 4 3 2 1 1 0!4! 1 4 3 2 1 Coefficiet of y 4 (or x 0 y 4 ) Note that each expressio has 4! i the umerator, with factorials i the deomiator correspodig to the expoets o x ad y. The Biomial Theorem We ow summarize these ideas i the iomial theorem. The Biomial Theorem I the expasio of (x y) for a positive iteger, there are 1 terms, give y the followig formula: (x y)! x! x 1! y x 2 y 2!0! ( 1)! 1! ( 2)! 2!...! y 0!! r The otatio is ofte used i place of ( Usig this otatio, we write the expasio as 0 1! r)!r! i the iomial expasio. (x y) x x1 y x 2 y 2... y. 2
654 (12-28) Chapter 12 Sequeces ad Series! Aother otatio for is ( r)!r! C r. Usig this otatio, we have (x y) C 0 x C 1 x 1 y C 2 x 2 y 2... C y. E X A M P L E 1 Usig the iomial theorem Write out the first three terms of (x y) 9. Solutio (x y) 9 x 9 x 8 y x 7 y 2 9!0! 8!1! 7!2!... x 9 9x 8 y 36x 7 y 2... E X A M P L E 2 Usig the iomial theorem Write the iomial expasio for (x 2 2a) 5. Solutio We expad a differece y writig it as a sum ad usig the iomial theorem: (x 2 2a) 5 (x 2 (2a)) 5 (x 2 ) 5 (x 2 ) 4 (2a) 1 (x 2 ) 3 (2a) 2 (x 2 ) 2 (2a) 3 5!0! 4!1! 3!2! 2!3! (x 2 )(2a) 4 (2a) 5 1!4! 0! x 10 10x 8 a 40x 6 a 2 80x 4 a 3 80x 2 a 4 32a 5 E X A M P L E 3 calculator close-up Because C! r, we ( r)! r! have 12 C 9 12! 12! ad 3! 12 C 3. 9! 3! So there is more tha oe way to compute 12!( 3!): Fidig a specific term Fid the fourth term of the expasio of (a ) 12. Solutio The variales i the first term are a 12 0, those i the secod term are a 11 1, those i the third term are a 10 2, ad those i the fourth term are a 9 3. So The fourth term is 220a 9 3. 1 2! a 9 3 220a 9 3. 3! Usig the ideas of Example 3, we ca write a formula for ay term of a iomial expasio. Formula for the kth Term of (x y) For k ragig from 1 to 1, the kth term of the expasio of (x y) is give y the formula! x k1 y k1. ( k 1)!(k 1)!
12.5 Biomial Expasios (12-29) 655 E X A M P L E 4 Fidig a specific term Fid the sixth term of the expasio of (a 2 2) 7. Solutio Use the formula for the kth term with k 6 ad 7: 7! (a 2 ) 2 (2) 5 21a 4 (32 5 ) 672a 4 5 (7 6 1)!(6 1)! We ca thik of the iomial expasio as a fiite series. Usig summatio otatio, we ca write the iomial theorem as follows. The Biomial Theorem (Usig Summatio Notatio) For ay positive iteger, (x y) i0! x i y i or (x y) ( i)!i! i0 xi y i. i E X A M P L E 5 Usig summatio otatio Write (a ) 5 usig summatio otatio. Solutio Use 5 i the iomial theorem: (a ) 5 5 i0 a 5i i (5 i)!i! WARM-UPS True or false? Explai your aswer. 1. There are 12 terms i the expasio of (a ) 12. False 2. The seveth term of (a ) 12 is a multiple of a 5 7. False 3. For all values of x, (x 2) 5 x 5 32. False 4. I the expasio of (x 5) 8 the sigs of the terms alterate. True 5. The eighth lie of Pascal s triagle is 18285670562881. True 6. The sum of the coefficiets i the expasio of (a ) 4 is 2 4. True 7. (a ) 3 3 3! a 3i i True i0 (3 i)! i! 8. The sum of the coefficiets i the expasio of (a ) is 2. True 9. 0! 1! True 7! 10. 21 True 5!2!
656 (12-30) Chapter 12 Sequeces ad Series 12. 5 EXERCISES Readig ad Writig After readig this sectio, write out the aswers to these questios. Use complete seteces. 1. What is a iomial expasio? The sum otaied for a power of a iomial is called a iomial expasio. 2. What is Pascal s triagle ad how do you make it? Pascal s triagle gives the coefficiets for (a ) for 1, 2, 3, ad so o. Each row starts ad eds with a 1. The other terms are otaied y addig the closest two terms i the precedig row. 3. What does! mea? The expressio! is the product of the positive itegers from 1 through. 4. What is the iomial theorem? The iomial theorem gives the expasio of (a ). Evaluate each expressio. 6! 5. 10 6. 6 2!3! 5!1! 8! 7. 56 8. 36 3! 2!7! Use the iomial theorem to expad each iomial. See Examples 1 ad 2. 9. (r t) 5 r 5 5r 4 t 10r 3 t 2 10r 2 t 3 5rt 4 t 5 10. (r t) 6 r 6 6r 5 t 15r 4 t 2 20r 3 t 3 15r 2 t 4 6rt 5 t 6 11. (m ) 3 m 3 3m 2 3m 2 3 12. (m ) 4 m 4 4m 3 6m 2 2 4m 3 4 13. (x 2a) 3 x 3 6ax 2 12a 2 x 8a 3 14. (a 3) 4 a 4 12a 3 54a 2 2 108a 3 81 4 15. (x 2 2) 4 x 8 8x 6 24x 4 32x 2 16 16. (x 2 a 2 ) 5 x 10 5a 2 x 8 10a 4 x 6 10a 6 x 4 5a 8 x 2 a 10 17. (x 1) 7 x 7 7x 6 21x 5 35x 4 35x 3 21x 2 7x 1 18. (x 1) 6 x 6 6x 5 15x 4 20x 3 15x 2 6x 1 Write out the first four terms i the expasio of each iomial. See Examples 1 ad 2. 19. (a 3) 12 a 12 36a 11 594a 10 2 5940a 9 3 20. (x 2y) 10 x 10 20x 9 y 180x 8 y 2 960x 7 y 3 21. (x 2 5) 9 x 18 45x 16 900x 14 10,500x 12 22. (x 2 1) 20 x 40 20x 38 190x 36 1140x 34 23. (x 1) 22 x 22 22x 21 231x 20 1540x 19 24. (2x 1) 8 256x 8 1024x 7 1792x 6 1792x 5 10 25. x y 3 2 1 26. a 2 5 8 a 2 x 10 5 9 x y 5 x 8 2 y 024 768 25 6 5 x 7 3 y 1 44 8 a 7 7 a 6 2 56 80 40 0 7 a 5 3 5 00 Fid the idicated term of the iomial expasio. See Examples 3 ad 4. 27. (a w) 13, 6th term 28. (m ) 12, 7th term 1287a 8 w 5 924m 6 6 29. (m ) 16, 8th term 30. (a ) 14, 6th term 11,440m 9 7 2002a 9 5 31. (x 2y) 8, 4th term 32. (3a ) 7, 4th term 448x 5 y 3 2835a 4 3 33. (2a 2 ) 20, 7th term 34. (a 2 w 2 ) 12, 5th term 635,043,840a 28 6 495a 16 w 8 Write each expasio usig summatio otatio. See Example 5. 35. (a m) 8 36. (z w) 13 8 i0 8! a 8i m i 13 1 3! z 13i w i (8 i)!i! (13 i)! i! i0 37. (a 2x) 5 38. (w 3m) 7 5 i0 ( (5 a 2) i 5i i)! i! xi 7 i 7! (3) w 7i m i (7 i)! i! i0 GETTING MORE INVOLVED 39. Discussio. Fid the triomial expasio for (a c) 3 y usig x a ad y c i the iomial theorem. a 3 3 c 3 3a 2 3a 2 c 3a 2 3ac 2 3 2 c 3c 2 6ac 40. Discussio. What prolem do you ecouter whe tryig to fid the fourth term i the iomial expasio for (x y) 120? How ca you overcome this prolem? Fid the fifth term i the iomial expasio for (x 2y) 100. 280,840x 117 y 3, 62,739,600x 96 y 4 COLLABORATIVE ACTIVITIES Lotteries Are Series(ous) Roerto ad his rother-i-law Horatio each have a child who will e graduatig from high school i 5 years. Each would like to uy his child a car for a graduatio preset. Horatio decides to uy two lottery tickets each week for the ext 5 years, hopig to wi ad uy a ew car for his child. The lottery tickets are $2.00 each at the local coveiece store. Roerto, who does t Groupig: Two to four studets per group Topic: Sequeces ad series elieve i lotteries, decides to set aside $4.00 each week ad to deposit this moey i a savigs accout each quarter for the ext 5 years to uy a used car. He fids a ak that will pay 5% yearly iterest compouded quarterly.