The Binomial Theorem

1. The Binomial Theorem 11.4 Derive the binomial theorem Use the binomial theorem Apply Pascal’s triangle

2. Binomial Theorem Expanding expressions in the form(a + b)n, where n is a natural number. Expressions occur in statistics, finite mathematics, computer science, and calculus. Combinations play a central role.

3. Binomial Theorem For any positive integer n and numbers a and b,

4. Binomial Theorem Since the combination formula can be used to evaluate binomial coefficients.

5. Use the binomial theorem to expand the expression (2a + 1)5. Solution Example: Applying the binomial theorem

6. Example: Applying the binomial theorem

7. Expanding (a + b)n Expanding (a+b)nfor increasing values of n gives the following results.

8. Pascal’s Triangle Used to efficiently compute the binomial coefficients C(n,r). The triangle consists of ones along the sides. Each element inside the triangle is the sum of the two numbers above it. It can be extended to include as many rows as needed.

9. Pascal’s Triangle

10. Expand each of the following. (a) (2x + 1)5 (b) (3x – y)3 Solution To expand (2x +1)5 , let a = 2x and b = 1 in the binomial theorem. We can use the sixth row of Pascal's triangle to obtain the coefficients 1, 5, 10,10, 5, and 1. Example: Expanding expressions with Pascal’s triangle

11. Let a = 3x and b = –y in the binomial theorem. Use the fourth row of Pascal's triangle to obtain the coefficients 1, 3, 3, and 1. Example: Expanding expressions with Pascal’s triangle

12. Finding the kth Term The binomial theorem gives all the terms of (a + b)n. An individual term can be found by noting that the (r + 1)st term in the binomial expansion for (a + b)n is given by the formula

13. Find the third term of (x – y)5. Solution Substituting the values for r = 2, n = 5,a = x, and b = –y in the formula for the(r + 1)st term yields Example: Finding the kth term in a binomial expansion