1. Copyright © 2007 Pearson Education, Inc. Slide 8-1

2. Copyright © 2007 Pearson Education, Inc. Slide 8-2 Chapter 8: Further Topics in Algebra 8.1Sequences and Series 8.2Arithmetic Sequences and Series 8.3Geometric Sequences and Series 8.4The Binomial Theorem 8.5Mathematical Induction 8.6Counting Theory 8.7Probability

3. Copyright © 2007 Pearson Education, Inc. Slide 8-3 8.4 The Binomial Theorem The binomial expansions reveal a pattern.

4. Copyright © 2007 Pearson Education, Inc. Slide 8-4 8.4 A Binomial Expansion Pattern The expansion of (x + y) n begins with x n and ends with y n. The variables in the terms after x n follow the pattern x n-1 y, x n-2 y 2, x n-3 y 3 and so on to y n. With each term the exponent on x decreases by 1 and the exponent on y increases by 1. In each term, the sum of the exponents on x and y is always n. The coefficients of the expansion follow Pascal’s triangle.

5. Copyright © 2007 Pearson Education, Inc. Slide 8-5 8.4 A Binomial Expansion Pattern Pascal’s Triangle Row

6. Copyright © 2007 Pearson Education, Inc. Slide 8-6 8.4 Pascal’s Triangle Each row of the triangle begins with a 1 and ends with a 1. Each number in the triangle that is not a 1 is the sum of the two numbers directly above it (one to the right and one to the left.) Numbering the rows of the triangle 0, 1, 2, … starting at the top, the numbers in row n are the coefficients of x n, x n-1 y, x n-2 y 2, x n-3 y 3, … y n in the expansion of (x + y) n.

7. Copyright © 2007 Pearson Education, Inc. Slide 8-7 8.4 n-Factorial n-Factorial For any positive integer n, and Example Evaluate (a) 5! (b) 7! Solution (a) (b)

8. Copyright © 2007 Pearson Education, Inc. Slide 8-8 8.4 Binomial Coefficients Binomial Coefficient For nonnegative integers n and r, with r < n,

9. Copyright © 2007 Pearson Education, Inc. Slide 8-9 8.4 Binomial Coefficients The symbols and for the binomial coefficients are read “n choose r” The values of are the values in the nth row of Pascal’s triangle. So is the first number in the third row and is the third.

10. Copyright © 2007 Pearson Education, Inc. Slide 8-10 8.4 Evaluating Binomial Coefficients Example Evaluate (a) (b) Solution (a) (b)

11. Copyright © 2007 Pearson Education, Inc. Slide 8-11 8.4 The Binomial Theorem Binomial Theorem For any positive integers n,

12. Copyright © 2007 Pearson Education, Inc. Slide 8-12 8.4 Applying the Binomial Theorem Example Write the binomial expansion of. Solution Use the binomial theorem

13. Copyright © 2007 Pearson Education, Inc. Slide 8-13 8.4 Applying the Binomial Theorem

14. Copyright © 2007 Pearson Education, Inc. Slide 8-14 8.4 Applying the Binomial Theorem Example Expand. Solution Use the binomial theorem with and n = 5,

15. Copyright © 2007 Pearson Education, Inc. Slide 8-15 8.4 Applying the Binomial Theorem Solution

16. Copyright © 2007 Pearson Education, Inc. Slide 8-16 8.4 rth Term of a Binomial Expansion rth Term of the Binomial Expansion The rth term of the binomial expansion of (x + y) n, where n > r – 1, is

17. Copyright © 2007 Pearson Education, Inc. Slide 8-17 8.4 Finding a Specific Term of a Binomial Expansion. Example Find the fourth term of. Solution Using n = 10, r = 4, x = a, y = 2b in the formula, we find the fourth term is