Binomial Theorem and Pascal’s Triangle

1. Binomial Theorem and Pascal’s Triangle PowerPoint was originally taken from the sight below and adapted To CCGPS by Tracy Bledsoe Unit #5 Binomial Theorem - GHP.ppt - Nunamaker nunamaker.wikispaces.com/.../Unit%20%235%20Binomial%20Theorem...

2. ObjectiveThe Student will be able to: • Use Pascal’s Triangle and the Binomial Theorem to expand a binomial raised to a power. • MCC9-12.A.APR.5

3. Essential Question: • How do I use Pascal’s Triangle and the Binomial Theorem in binomial expansion?

4. Pascal’s Triangle

5. Binomial Theorem

7. Binomial Theorem • Notice each expression has n + 1 terms • The degree of each term is equal to n • The exponent of each a decreases by 1 and the exponent of each b increases by 1 for each succeeding term in the series • The coefficients come from Pascal’s Triangle • In subtraction alternate signs starting with positive then negative

8. Expand using the Binomial Theorem and Pascal’s Triangle

9. Answers to 1, 2, and 3 part A • 1. x4 + 4x3y + 6x2y2 + 4xy3 + y4 • 2. a5 + 5a4b – 10a3b2 + 10a2b3 – 5ab4 + b5 • 3. m6 + 6m5n – 15m4n2 + 20m3n3 - 15m2n4 + 6mn5 – n6

10. Binomial Theorem • Write the general rule for the binomial using Pascal’s Triangle • Substitute into the general rule • Simplify your expression

11. Expand using the Binomial Theorem and Pascal’s Triangle

12. Answers to 1, 2, and 3 part B • 1. 16x4 + 96x3 + 216x2 + 216x + 81 • 2. 243a10 + 810a8b + 1080a6b2 + 720a4b3 + 240a2b4 + 32b5 • 3. 729m6 + 5832m5n – 19440m4n2 + 34560m3n3 – 34560m2n4 + 18432mn5 – 4096n6

13. Use the previous term method to determine each of the following

14. Answers to 1 and 2 Part C • 1. 128x7 + 448x6y3 – 672x5y6 + 560x4y9 • 2. 729a6 + 384a5b2 + 4860a4b4 + 4320a3b6

15. Factorial If n > 0 is an integer, the factorial symbol n! is defined as follows: 0! = 1 and 1! = 1 n! = n(n – 1) •…• 3 • 2 • 1 if n > 2 4! = 4 • 3 • 2 • 1 = 24 6! = 6 • 5 • 4 • 3 • 2 • 1 = 720

16. Factorial

17. Evaluate the following expressions

18. Answers to 1 – 6 Part D • 1. 4 2. 15 • 3. 9 4. 10 • 5. 35 6. 36

19. We can use the Binomial Theorem to find a particular term in an expression without writing the entire expansion.

20. Find the stated term.

21. Answers to 1, 2, and 3 Part E • 1. 5376x3 • 2. -21504x2 • 3. 314,928x4

23. Answers to 1, 2, and 3 Part F • 1. 103,680 • 2. 25,344 • 3. 53,760

24. Use polynomial identities to solve problems. • Know and apply that the Binomial Theorem gives the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)

25. I can . . . • Use Pascal’s Triangle and the Binomial Theorem to expand binomials.