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5-Minute Check Lesson 12-6A

5-Minute Check Lesson 12-6A. Math ador Gameplan. Section 12.6: Binomial Theorem CA Standards: MA 5.2 Daily Objective ( 5/30/13 ): Students will be able to apply the Binomial Theorem to expand binomials Homework: page 804 (# 12 to 32) . Problems. What is (x+y) 0 ? 1

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5-Minute Check Lesson 12-6A

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  1. 5-Minute Check Lesson 12-6A

  2. MathadorGameplan Section 12.6:Binomial Theorem CA Standards: MA 5.2 Daily Objective (5/30/13): Students will be able to apply the Binomial Theorem to expand binomials Homework:page 804 (#12 to 32)

  3. Problems • What is (x+y)0? • 1 • What is (x+y)1? • x+y • What is (x+y)2? • x2+2xy+y2 • What is (x+y)3? • x3+3x2y+3xy2+y3 • What is (x+y)4? • Let’s use the binomial theorem

  4. Patterns • Expanding (x + y)n • There are n + 1 terms • The first term is xn and the last term is yn • In successive terms the exponent of x decreases by 1 and the exponent of y increases by 1 • The degree of each term is n • The coefficients are symmetric

  5. Pascal’s Triangle (x+y)n • 1 • x+y • x2+2xy+y2 • x3+3x2y+3xy2+y3 • n=0 • n=1 • n=2 • n=3 • n=4 • n=5 • n=6

  6. Lesson Overview 12-6A

  7. Problem • Now expand (x + y)4 • x4 + 4x3y + 6x2y2 + 4xy3 + y4

  8. Practice – Use the binomial theorem to expand the following • (a + 3)6 • (5 – y)3 • (3x + 1)4 Answers • a6 + 18a5 + 135a4 + 540a3 + 1215a2 + 1458a + 729 • 125 – 75y + 15y2 – y3 • 81x4 + 108x3 + 54x2 + 12x + 1

  9. Definition of a Binomial Coefficient “n above r” For nonnegative integers n and r, with n > r, the expression is called a binomial coefficient and is defined by

  10. Example • Evaluate Solution:

  11. Practice • Evaluate the following • a) b) c) d)

  12. Answers • Evaluate the following • a) 20 b) 1 c) 28 d) 1

  13. The Binomial Theorem For any positive integer n

  14. Example • Expand Solution:

  15. Practice • Expand (x – y)9 Answer x9 - 9x8y + 36x7y2 - 84x6y3 + 126x5y4 - 126x4y5 + 84x3y6 - 36x2y7 + 9xy8 - y9

  16. Finding a Particular Term in a Binomial Expansion • The rth term of the expansion of (a+b)n is

  17. Example: Find the fourth term in the expansion of (3x + 2y)7. Solution We will use the formula for the rth term of the expansion (a+b)n, to find the fourth term of (3x+ 2y)7. For the fourth term of (3x+ 2y)7, n= 7, r= 4 – 1, a= 3x, and b= 2y. Thus, the fourth term is

  18. Practice • Find the fifth term of (2x + y)9 • Find the sixth term of (a – b)7 • Find the fourth term of (2x – 3y)6 Answers 4032x5y4 -21a2b5 -4320x3y3

  19. Permutations and Combinations • Permutation: The arrangements of items in a certain order • Combination: A group of items in NO certain order

  20. Permutations • The number of possible permutations if r items are taken from n items is

  21. Combinations • The number of possible combinations if r items are taken from n items is

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