1 / 21

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

eshana
Télécharger la présentation

5-Minute Check Lesson 12-6A

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related