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MS-Sol-EE-C02 - CHAPTER 2 THE BINOMIAL EXPANSION EXERCISE...

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CHAPTER 2 THE BINOMIAL EXPANSION EXERCISE 2.1 Section 2.1 The binomial theorem (page 33) 1. ( a + b ) 3 = a 3 + 3 a 2 b + 3 ab 2 + b 3 2. (1 + x ) 7 = 1 + 7 x + 21 x 2 + 35 x 3 + 35 x 4 + 21 x 5 + 7 x 6 + x 7 3. ( a + b ) 8 = a 8 + 8 a 7 b + 28 a 6 b 2 + 56 a 5 b 3 + 70 a 4 b 4 + 56 a 3 b 5 + 28 a 2 b 6 + 8 ab 7 + b 8 4. (1 + x ) 9 = 1 + 9 x + 36 x 2 + 84 x 3 + 126 x 4 + 126 x 5 + 84 x 6 + 36 x 7 + 9 x 8 + x 9 5. (1 + 2 x ) 2 = 1 + 2(2 x ) + (2 x ) 2 = 1 + 4 x + 4 x 2 6. (1 + 6 x ) 3 = 1 + 3(6 x ) + 3(6 x ) 2 + (6 x ) 3 = 1 + 18 x + 108 x 2 + 216 x 3 7. (1 - 4 x ) 4 = 1 - 4(4 x ) + 6(4 x ) 2 - 4(4 x ) 3 + (4 x ) 4 = 1 - 16 x + 96 x 2 - 256 x 3 + 256 x 4 8. (2 x + y ) 5 = (2 x ) 5 + 5(2 x ) 4 y + 10(2 x ) 3 y 2 + 10(2 x ) 2 y 3 + 5(2 x ) y 4 + y 5 = 32 x 5 + 80 x 4 y + 80 x 3 y 2 + 40 x 2 y 3 + 10 xy 4 + y 5 9. ( a + 3 b ) 6 = a 6 + 6 a 5 (3 b ) + 15 a 4 (3 b ) 2 + 20 a 3 (3 b ) 3 + 15 a 2 (3 b ) 4 + 6 a (3 b ) 5 + (3 b ) 6 = a 6 + 18 a 5 b + 135 a 4 b 2 + 540 a 3 b 3 + 1 215 a 2 b 4 + 1 458 ab 5 + 729 b 6 10. (1 - 2 x 2 ) 7 = 1 - 7(2 x 2 ) + 21(2 x 2 ) 2 - 35(2 x 2 ) 3 + 35(2 x 2 ) 4 - 21(2 x 2 ) 5 + 7(2 x 2 ) 6 - (2 x 2 ) 7 = 1 - 14 x 2 + 84 x 4 - 280 x 6 + 560 x 8 - 672 x 10 + 448 x 12 - 128 x 14 11. 5 1 - x x = 5 4 3 2 2 3 4 5 1 1 5 + 1 10 1 10 + 1 5 - - - x x x x x x x x x x = 5 3 3 5 1 5 + 10 10 + 5 x x x x x x - - - 12. 4 2 1 + x x = x 4 + 4 x 3 2 1 x + 6 x 2 2 2 1 x + 4 x 3 2 1 x + 4 2 1 x = x 4 + 4 x + 2 6 x + 5 4 x + 8 1 x 13. (1 + 3 x ) 10 = 1 + 3 10 3 2 10 2 10 1 ) 3 ( + ) 3 ( + ) 3 ( x C x C x C + . . . = 1 + 30 x + 405 x 2 + 3 240 x 3 + . . . 14. (1 - 2 x ) 17 = 1 - C 17 1 (2 x ) + C 17 2 (2 x ) 2 - C 17 3 (2 x ) 3 + . . . = 1 - 34 x + 544 x 2 - 5 440 x 3 + . . . 15
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C HAPTER 2 T HE B INOMIAL E XPANSION 15. (2 - x ) 9 = 3 6 9 3 2 7 9 2 8 9 1 9 ) 2 ( ) 2 ( + ) 2 ( 2 x C x C x C - - + . . . = 512 - 2 304 x + 4 608 x 2 - 5 376 x 3 + . . . 16. (3 + x ) 11 = 3 11 + C 11 1 (3 10 ) x + C 11 2 (3 9 ) x 2 + C 11 3 (3 8 ) x 3 + . . . = 177 147 + 649 539 x + 1 082 565 x 2 + 1 082 565 x 3 + . . . 17. (1 + x ) 6 (1 - x ) 4 = (1 + 6 x + 15 x 2 + 20 x 3 + . . . ) (1 - 4 x + 6 x 2 - 4 x 3 + x 4 ) = 1 + 2 x - 3 x 2 - 8 x 3 + . . . 18. (1 + 3 x ) 2 (1 + x ) 5 = [1 + 2(3 x ) + (3 x ) 2 ] (1 + 5 x + 10 x 2 + 10 x 3 + . . .) = (1 + 6 x + 9 x 2 ) (1 + 5 x + 10 x 2 + 10 x 3 + . . .) = 1 + 11 x + 49 x 2 + 115 x 3 + . . . 19. (3 + x ) 5 (2 + 7 x ) 7 = [3 5 + 5(3 4 ) x + 10(3 3 ) x 2 + 10(3 2 ) x 3 + . . .] [2 7 + 7(2 6 ) (7 x ) + 21(2 5 ) (7 x ) 2 + 35(2 4 ) (7 x ) 3 + . . . ] = [243 + 405 x + 270 x 2 + 90 x 3 + . . . ] [128 + 3 136 x + 32 928 x 2 + 192 080 x 3 + . . . ] = 31 104 + 813 888 x + 9 306 144 x 2 + 60 869 520 x 3 + . . . 20. (2 - x ) 3 (5 + x ) 4 = [2 3 - 3(2 2 ) x + 3(2) x 2 - x 3 ] [5 4 + 4(5 3 ) x + 6(5 2 ) x 2 + 4(5) x 3 + x 4 ] = (8 - 12 x + 6 x 2 - x 3 ) (625 + 500 x + 150 x 2 + 20 x 3 + x 4 ) = 5 000 - 3 500 x - 1 050 x 2 + 735 x 3 + . . . 21. (1 + x - 2 x 2 ) 5 = [1 + x (1 - 2 x )] 5 = 1 + 5 x (1 - 2 x ) + 10 x 2 (1 - 2 x ) 2 + 10 x 3 (1 - 2 x ) 3 + . . . = 1 + 5 x - 10 x 2 + 10 x 2 (1 - 4 x + 4 x 2 ) + 10 x 3 + . . . = 1 + 5 x - 30 x 3 + . . . 22. (1 + 3 x + x 2 ) 8 = [1 + x (3 + x )] 8 = 1 + 8 x (3 + x ) + 28 x 2 (3 + x ) 2 + 56 x 3 (3 + x ) 3 + . . . = 1 + 24 x + 8 x 2 + 28 x 2 (9 + 6 x + x 2 ) + 1 512 x 3 + . . . = 1 + 24 x + 260 x 2 + 1 680 x 3 + . . . 23. (1 - 4 x - 3 x 2 ) 12 = [1 - x (4 + 3 x )] 12 = 1 - 12 x (4 + 3 x ) + 66 x 2 (4 + 3 x ) 2 - 220 x 3 (4 + 3 x ) 3 + . . . = 1 - 48 x - 36 x 2 + 66 x 2 (16 + 24 x + 9 x 2 ) - 14 080 x 3 + . . . = 1 - 48 x + 1 020 x 2 - 12 496 x 3 + . . . 24. (1 - x + 5 x 2 ) 11 = [1 - x (1 - 5 x )] 11 = 1 - 11 x (1 - 5 x ) + 55 x 2 (1 - 5 x ) 2 - 165 x 3 (1 - 5 x ) 3 + . . . = 1 - 11 x + 55 x 2 + 55 x 2 (1 - 10 x + 25 x 2 ) - 165 x 3 + . . . = 1 - 11 x + 110 x 2 - 715 x 3 + . . . 25. The ( r + 1)th term = C x x r r r 9 9 2 2 3 ( ) - - = C x r r r r 9 9 9 3 2 3 ( ) ( ) - - - If the term is independent of x , then 9 - 3 r = 0 r = 3 The term independent of x = C 3 9 9 3 3 2 3 ( ) ( ) - - = - 145 152 16
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S ECTION 2.1 T HE B INOMIAL T HEOREM 26. The ( r + 1)th term = C x x r r r 8 8 3 4 - = C x r r r 8 8 4 4 - It is a constant term if 8 - 4 r = 0 r = 2 The constant term= C 2 8 4 2 = 448 27. The 5th term = C 10 4 x 10–4 ( - 2 y ) 4 = 3 360 x 6 y 4 28. The 7th term = C x y 6 11 11 6 6 2 3
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