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1114 ch 11 day 14

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• 1114 ch 11 day 14

1. 1. 11.6 The Binomial TheoremMatthew 7:7 "Ask, and it shall be given you; seek, and yeshall ﬁnd; knock, and it shall be opened unto you."
2. 2. How do we expand ( a + b ) ? n
3. 3. How do we expand ( a + b ) ? nWe will explore two methods:
4. 4. How do we expand ( a + b ) ? nWe will explore two methods:1) Using Pascal’s Triangle (a recursive method)
5. 5. How do we expand ( a + b ) ? nWe will explore two methods:1) Using Pascal’s Triangle (a recursive method)2) Using the Binomial Theorem (an explicit method that uses combinatorics)
6. 6. Pascal’s Triangle
7. 7. Pascal’s Triangle Recall from last year ...
8. 8. Pascal’s Triangle Provides coefﬁcients for expansion of a binomial
9. 9. Pascal’s Triangle Provides coefﬁcients for expansion of a binomial 0 (a + b) = 1 1 (a + b) = 1a + 1b 2 2 2 (a + b) = 1a + 2ab + 1b 3 3 2 2 3 (a + b) = 1a + 3a b + 3ab + 1b 4 4 3 2 2 3 4 (a + b) = 1a + 4a b + 6a b + 4ab + 1b 5 5 4 3 2 2 3 4 5 (a + b) = 1a + 5a b + 10a b + 10a b + 5ab + 1b
10. 10. Pascal’s Triangle Provides coefﬁcients for expansion of a binomial 0 (a + b) = 1 1 (a + b) = 1a + 1b 2 2 2 (a + b) = 1a + 2ab + 1b 3 3 2 2 3 (a + b) = 1a + 3a b + 3ab + 1b 4 4 3 2 2 3 4 (a + b) = 1a + 4a b + 6a b + 4ab + 1b 5 5 4 3 2 2 3 4 5 (a + b) = 1a + 5a b + 10a b + 10a b + 5ab + 1b Also notice the pattern of the exponents!
11. 11. Pascal’s Triangle Provides coefﬁcients for expansion of a binomial 0 (a + b) = 1 1 (a + b) = 1a + 1b 2 2 2 (a + b) = 1a + 2ab + 1b 3 3 2 2 3 (a + b) = 1a + 3a b + 3ab + 1b 4 4 3 2 2 3 4 (a + b) = 1a + 4a b + 6a b + 4ab + 1b 5 5 4 3 2 2 3 4 5 (a + b) = 1a + 5a b + 10a b + 10a b + 5ab + 1b Also notice the pattern of the exponents! Let’s expand ( a + b ) . 6 (do on the board)
12. 12. Let’s expand ( 3 − xy ) 4
13. 13. Let’s expand ( 3 − xy ) 4First, write out ( a + b ) in expanded form. 4
14. 14. Let’s expand ( 3 − xy ) 4First, write out ( a + b ) in expanded form. 4 4 3 2 2 3 4 a + 4a b + 6a b + 4ab + b
15. 15. Let’s expand ( 3 − xy ) 4First, write out ( a + b ) in expanded form. 4 4 3 2 2 3 4 a + 4a b + 6a b + 4ab + bThen substitute 3 for a and -xy for b
16. 16. Let’s expand ( 3 − xy ) 4First, write out ( a + b ) in expanded form. 4 4 3 2 2 3 4 a + 4a b + 6a b + 4ab + bThen substitute 3 for a and -xy for b 4 3 2 2 3 4 ( 3) + 4 ( 3) ( −xy ) + 6 ( 3) ( −xy ) + 4 ( 3) ( −xy ) + ( −xy )
17. 17. Let’s expand ( 3 − xy ) 4First, write out ( a + b ) in expanded form. 4 4 3 2 2 3 4 a + 4a b + 6a b + 4ab + bThen substitute 3 for a and -xy for b 4 3 2 2 3 4 ( 3) + 4 ( 3) ( −xy ) + 6 ( 3) ( −xy ) + 4 ( 3) ( −xy ) + ( −xy )and simplify
18. 18. Let’s expand ( 3 − xy ) 4First, write out ( a + b ) in expanded form. 4 4 3 2 2 3 4 a + 4a b + 6a b + 4ab + bThen substitute 3 for a and -xy for b 4 3 2 2 3 4 ( 3) + 4 ( 3) ( −xy ) + 6 ( 3) ( −xy ) + 4 ( 3) ( −xy ) + ( −xy )and simplify 2 2 3 3 4 4 81− 108xy + 54x y − 12x y + x y
19. 19. Pascal’s Triangle works great when n issmall in ( a + b ) nThe Binomial Theorem is better when nis large. Let’s take a look at that now.
20. 20. The Binomial Theorem
21. 21. The Binomial Theorem a, b ∈ ° ; n ∈ positive integers n ⎛ n ⎞ n ⎛ n ⎞ n−1 ⎛ n ⎞ n−2 2 ⎛ n ⎞ n−1 ⎛ n ⎞ n(a + b) = ⎜ ⎟ a + ⎜ ⎟ a b + ⎜ ⎟ a b +K + ⎜ ⎟ ab + ⎜ n ⎟ b ⎝ 0 ⎠ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ n − 1⎠ ⎝ ⎠
22. 22. The Binomial Theorem a, b ∈ ° ; n ∈ positive integers n ⎛ n ⎞ n ⎛ n ⎞ n−1 ⎛ n ⎞ n−2 2 ⎛ n ⎞ n−1 ⎛ n ⎞ n(a + b) = ⎜ ⎟ a + ⎜ ⎟ a b + ⎜ ⎟ a b +K + ⎜ ⎟ ab + ⎜ n ⎟ b ⎝ 0 ⎠ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ n − 1⎠ ⎝ ⎠Let’s review combinations ... (next slide)
23. 23. Combinations ⎛ n ⎞ n! ⎜ r ⎟ = r!( n − r )! ⎝ ⎠
24. 24. Combinations ⎛ n ⎞ n! ⎜ r ⎟ = r!( n − r )! ⎝ ⎠Example: ⎛ 5 ⎞ 5! ⎜ 2 ⎟ = 2!( 5 − 2 )! ⎝ ⎠ 5 ⋅ 4 ⋅ 3⋅ 2 ⋅1 = ( 2 ⋅1)( 3⋅ 2 ⋅1) 5⋅4 = 2 = 10
25. 25. Combinations ⎛ n ⎞ n! ⎜ r ⎟ = r!( n − r )! ⎝ ⎠Example: ⎛ 5 ⎞ 5! ⎜ 2 ⎟ = 2!( 5 − 2 )! ⎝ ⎠ 5 ⋅ 4 ⋅ 3⋅ 2 ⋅1 = ( 2 ⋅1)( 3⋅ 2 ⋅1) 5⋅4 = 2 = 10Let’s review how your calculator can do this ...
26. 26. ⎛ 5 ⎞To do ⎜ ⎟ enter 5 nCr 2 ⎝ 2 ⎠
27. 27. ⎛ 5 ⎞To do ⎜ ⎟ enter 5 nCr 2 ⎝ 2 ⎠For all combinations, you can use your calculator.In your work, just show the combination notation,but not the nCr notation.
28. 28. 4Use the Binomial Theorem to expand ( a + b )
29. 29. 4Use the Binomial Theorem to expand ( a + b ) ⎛ 4 ⎞ 4 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 2 2 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 4 ⎜ 0 ⎟ a + ⎜ 1⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3 ⎟ ab + ⎜ 4 ⎟ b ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
30. 30. 4Use the Binomial Theorem to expand ( a + b ) ⎛ 4 ⎞ 4 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 2 2 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 4 ⎜ 0 ⎟ a + ⎜ 1⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3 ⎟ ab + ⎜ 4 ⎟ b ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 4 3 2 2 3 4 a + 4a b + 6a b + 4ab + b
31. 31. 4Use the Binomial Theorem to expand ( a + b ) ⎛ 4 ⎞ 4 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 2 2 ⎛ 4 ⎞ 3 ⎛ 4 ⎞ 4 ⎜ 0 ⎟ a + ⎜ 1⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3 ⎟ ab + ⎜ 4 ⎟ b ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 4 3 2 2 3 4 a + 4a b + 6a b + 4ab + b nFor more complex binomials, ﬁrst expand ( a + b )and then substitute in for a and b ... just like wedid with the Pascal’s Triangle method.
32. 32. 6Use the Binomial Theorem to expand ( x −2 )
33. 33. 6 Use the Binomial Theorem to expand ( x −2 )⎛ 6 ⎞ 6 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 4 2 ⎛ 6 ⎞ 3 3 ⎛ 6 ⎞ 2 4 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 6⎜ 0 ⎟ a + ⎜ 1 ⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3⎟ a b + ⎜ 4 ⎟ a b + ⎜ 5 ⎟ ab + ⎜ 6 ⎟ b⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
34. 34. 6 Use the Binomial Theorem to expand ( x −2 )⎛ 6 ⎞ 6 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 4 2 ⎛ 6 ⎞ 3 3 ⎛ 6 ⎞ 2 4 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 6⎜ 0 ⎟ a + ⎜ 1 ⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3⎟ a b + ⎜ 4 ⎟ a b + ⎜ 5 ⎟ ab + ⎜ 6 ⎟ b⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ a= x b = −2
35. 35. 6 Use the Binomial Theorem to expand ( x −2 )⎛ 6 ⎞ 6 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 4 2 ⎛ 6 ⎞ 3 3 ⎛ 6 ⎞ 2 4 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 6⎜ 0 ⎟ a + ⎜ 1 ⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3⎟ a b + ⎜ 4 ⎟ a b + ⎜ 5 ⎟ ab + ⎜ 6 ⎟ b⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ a= x b = −2 6 5 4 2 3 3 2 4 1 x + 6 x ( −2 ) + 15 x ( −2 ) + 20 x ( −2 ) + 15 x ( −2 ) 5 6 +6 x ( −2 ) + 1( −2 )
36. 36. 6 Use the Binomial Theorem to expand ( x −2 )⎛ 6 ⎞ 6 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 4 2 ⎛ 6 ⎞ 3 3 ⎛ 6 ⎞ 2 4 ⎛ 6 ⎞ 5 ⎛ 6 ⎞ 6⎜ 0 ⎟ a + ⎜ 1 ⎟ a b + ⎜ 2 ⎟ a b + ⎜ 3⎟ a b + ⎜ 4 ⎟ a b + ⎜ 5 ⎟ ab + ⎜ 6 ⎟ b⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ a= x b = −2 6 5 4 2 3 3 2 4 1 x + 6 x ( −2 ) + 15 x ( −2 ) + 20 x ( −2 ) + 15 x ( −2 ) 5 6 +6 x ( −2 ) + 1( −2 ) 5 3 1 3 2 x − 12x + 60x − 160x + 240x − 192x + 64 2 2 2
37. 37. HW #11“I am a little pencil in the hand of a writing God who issending a love letter to the world.” Mother Teresa