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# 1114 ch 11 day 14

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• 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. How do we expand ( a + b ) ? n
• 3. How do we expand ( a + b ) ? nWe will explore two methods:
• 4. How do we expand ( a + b ) ? nWe will explore two methods:1) Using Pascal’s Triangle (a recursive method)
• 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. Pascal’s Triangle
• 7. Pascal’s Triangle Recall from last year ...
• 8. Pascal’s Triangle Provides coefﬁcients for expansion of a binomial
• 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. 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. 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. Let’s expand ( 3 − xy ) 4
• 13. Let’s expand ( 3 − xy ) 4First, write out ( a + b ) in expanded form. 4
• 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. 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. 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. 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. 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. 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. The Binomial Theorem
• 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. 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. Combinations ⎛ n ⎞ n! ⎜ r ⎟ = r!( n − r )! ⎝ ⎠
• 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. 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. ⎛ 5 ⎞To do ⎜ ⎟ enter 5 nCr 2 ⎝ 2 ⎠
• 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. 4Use the Binomial Theorem to expand ( a + b )
• 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. 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. 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. 6Use the Binomial Theorem to expand ( x −2 )
• 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. 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. 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. 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. HW #11“I am a little pencil in the hand of a writing God who issending a love letter to the world.” Mother Teresa