10 5 The Binomial Theorem

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10 5 The Binomial Theorem Daily Outcomes: I can use Pascal's triangle to write binomial expansions I can use the Binomial Theorem to write and find the coefficients of specified terms in binomial

1 10 5 The Binomial Theorem Daily Outcomes: I can use Pascal's triangle to write binomial expansions I can use the Binomial Theorem to write and find the coefficients of specified terms in binomial expansions What would the binomial look like if the number of offspring was 10? For 10 offspring, how many would probably be male and how many would probably be female? Dec 31 12:26 PM Apr 25 11:39 AM 1

2 Pascal's Triangle The top row is called zeroth row because it corresponds with the binomial expansion (a + b) 0 Notice each row begins with 1 and ends with 1 Every other number is formed by adding the two numbers immediately above that number in the previous row Apr 8 6:45 PM Expansion of a binomial raised to an integer power Observe the following patterns in the expansions of (a + b) n Each expansion has n + 1 terms The first term is a n, and the last term is b n In successive terms, the powers of a decrease by 1 and the powers of b increase by 1 The sum of the exponents in each term is n The coefficients, in red above, increase and then decrease in a symmetric pattern Dec 31 12:26 PM 2

3 Apr 8 7:00 PM Use Pacal's Triangle to expand each binomial. 1. (a + b) 6 2. (3x + 2) 5 Dec 31 12:43 PM 3

4 3. (a + b) 8 4. (2x + 3y) 5 Apr 8 6:58 PM Shortcut to expanding a binomial 1. Decrease first term from exponent, exponent minus 1, exponent minus 2, etc until exponent 0 2. Start at the right and decrease second term from exponent, exponent minus 1, exponent minus 2, etc until exponent 0 3. Determine coefficient for each term: multiply exponent times coefficient, divide by the number of term Dec 31 12:43 PM 4

5 Expand each binomial. 5. (a + b) 6 6. (3x + 2) 5 7. (a + b) 8 8. (2x + 3y) 5 Apr 8 7:06 PM When the binomial is a difference, the expansion is the same except for the signs. Start with a positive and alternate negative, positive, etc, until all the terms are done. Expand the binomials. 9. (x 2y) (2x 7) (2x 3y) 4 Dec 31 2:38 PM 5

Binomial Theorem An explicit formula for computing each binomial coefficient is developed by considering (a + b) n as the product of n factors in which each factor contributes an a or a b to each

Because those factors that do not contribute a b will by default contribute an a, the number of ways to form the product a n r b r can be more simply thought of as the number of ways to choose r

is defined as the product of all positive integers from 1 to n, then: 1! = 1*1 = 1 2! = 1*2 = 2 3! = 1*2*3 = 6 4! = 1*2*3*4 = 24... n! = 1*2*3*...*(n 2)*(n 1)*n and so on. Logically, n!

6 Binomial Theorem An explicit formula for computing each binomial coefficient is developed by considering (a + b) n as the product of n factors in which each factor contributes an a or a b to each product in the expansion. Because those factors that do not contribute a b will by default contribute an a, the number of ways to form the product a n r b r can be more simply thought of as the number of ways to choose r factors to contribute a b to the product from the n factors available. This is the combination given by also written Dec 31 2:40 PM Proof of 0! Consider n! = n*(n 1)! If n! is defined as the product of all positive integers from 1 to n, then: 1! = 1*1 = 1 2! = 1*2 = 2 3! = 1*2*3 = 6 4! = 1*2*3*4 = n! = 1*2*3*...*(n 2)*(n 1)*n and so on. Logically, n! can also be expressed n*(n 1)!. Therefore, at n=1, using n! = n*(n 1)! 1! = 1*0! which simplifies to 1 = 0! Because (n+1)! = n! (n+1), and consequently n! = (n+1)!/(n+1) So we have 3! = 1x2x3 = 6 2! = 3!/3 = 6/3 = 2 1! = 2!/2 = 2/2 = 1 0! = 1!/1 = 1/1 = 1 ( 1)! = 0!/0 = 1/0 = undefined Rearranging you get: (n 1)! = n!/n If you were to plug in n=1, you can see from the above equation: 0! = 1!/1 or 0!=1 Feb 11 12:17 PM 6

7 Dec 31 2:38 PM Apr 8 7:35 PM 7

8 Example 3: Find the coefficient of the indicated term in each expansion. 12. (a b) 10, fourth term 13. (x + y) 9, sixth term 14. (a b) 13, third term Apr 8 7:34 PM Apr 8 7:47 PM 8

9 Example 4: Find the coefficient of the indicated term in each binomial expansion. 15. (3x 4y) 8, x 7 y term 16. (2x 3y) 8, x 3 y 5 term 17. (2p + 1) 15, 11th term Dec 31 3:15 PM Apr 8 7:53 PM 9

Example 5: 18. During a player's turn in a certain board game, players must spin a spinner. The four possible colors the spinner can land on are green, blue, red, or yellow.

10 Example 5: 18. During a player's turn in a certain board game, players must spin a spinner. The four possible colors the spinner can land on are green, blue, red, or yellow. If the probability for all four colors is equal, what is the probability of landing on green 5 times out of 10 spins? 19. A fair coin is flipped 8 times. Find the probability of each outcome. a. exactly 3 heads b. exactly 6 tails Apr 8 7:55 PM Dec 31 9:04 PM 10

11 Apr 8 8:04 PM Example 6: Use the binomial theorem to expand each binomial. 20. (2t + 3u) (a 2b 2 ) (5m + 4) (8x 2 2y) 6 Apr 8 8:01 PM 11

12 Apr 8 8:11 PM 12

5.9: The Binomial Theorem

5.9: The Binomial Theorem Pascal s Triangle 1. Show that zz = 1 + ii is a solution to the fourth degree polynomial equation zz 4 zz 3 + 3zz 2 4zz + 6 = 0. 2. Show that zz = 1 ii is a solution to the fourth