Monday: Announcements Progress Reports this Thursday 3 rd period Tuesday/Wednesday STARR Testing, so NO Tutorials (30 minute classes) Tuesday Periods 1,3,5,7 Wednesday Periods 2,4,6,8

The Binomial Theorem Section 9-5

3 Objectives Be able to expand binomials with expansion theorem Know Pascal’s triangle for finding coefficients. Find specific terms and coefficients in an expansion

4 Group Work Expand The following binomials: (x + y) 2 (x + y) 3 (x + y) 4 (x + y) 5 (x + y) 6

5 Binomial Expansions The binomial theorem provides a useful method for raising any binomial to a nonnegative integral power. Consider the patterns formed by expanding (x + y) n. (x + y) 0 = 1 (x + y) 1 = x + y (x + y) 2 = x 2 + 2xy + y 2 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 Notice that each expansion has n + 1 terms. 1 term 2 terms 3 terms 4 terms 5 terms 6 terms Example: (x + y) 10 will have , or 11 terms.

6 Patterns of Exponents in Binomial Expansions Consider the patterns formed by expanding (x + y) n. (x + y) 0 = 1 (x + y) 1 = x + y (x + y) 2 = x 2 + 2xy + y 2 (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x + y) 4 = x 4 + 4x 3 y + 6x 2 y 2 + 4xy 3 + y 4 (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 1. The exponents on x decrease from n to 0. The exponents on y increase from 0 to n. 2. Each term is of degree n. Example: The 5 th term of (x + y) 10 is a term with x 6 y 4.”

7 Binomial Coefficients The coefficients of the binomial expansion are called binomial coefficients. The coefficients have symmetry. The coefficient of x n–r y r in the expansion of (x + y) n is written or n C r. (x + y) 5 = x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy 4 + y 5 The first and last coefficients are 1. The coefficients of the second and second to last terms are equal to n. 1 1 Example: What are the last 2 terms of (x + y) 10 ? Since n = 10, the last two terms are 10xy 9 + 1y 10. So, the last two terms of (x + y) 10 can be expressed as 10 C 9 xy C 10 y 10 or as xy 9 + y 10.

8 The triangular arrangement of numbers below is called Pascal’s Triangle. Each number in the interior of the triangle is the sum of the two numbers immediately above it. The numbers in the n th row of Pascal’s Triangle are the binomial coefficients for (x + y) n. 1 1 st row nd row rd row th row th row 0 th row = = 3

9 Example: Use Pascal’s Triangle to expand (2a + b) 4. (2a + b) 4 = 1(2a) 4 + 4(2a) 3 b + 6(2a) 2 b 2 + 4(2a)b 3 + 1b 4 = 1(16a 4 ) + 4(8a 3 )b + 6(4a 2 b 2 ) + 4(2a)b 3 + b 4 = 16a a 3 b + 24a 2 b 2 + 8ab 3 + b 4 Example: Pascal’s Triangle 1 1 st row nd row rd row th row 0 th row1

10 The symbol n! (n factorial) denotes the product of the first n positive integers. 0! is defined to be 1. n! = n(n – 1)(n – 2)  ! = 1 4! = = 24 6! = = 720 Formula for Binomial Coefficients For all nonnegative integers n and r, Example: Formula for the Binomial Coefficients

11 Example: Use the formula to calculate the binomial coefficients 10 C 5, 15 C 0, and. Example: Binomial coefficients

12 Binomial Theorem Example: Use the Binomial Theorem to expand (x 4 + 2) 3. Definition: Binomial Theorem

13 Although the Binomial Theorem is stated for a binomial which is a sum of terms, it can also be used to expand a difference of terms. Simply rewrite (x + y) n as (x + (– y)) n and apply the theorem to this sum. Example: Use the Binomial Theorem to expand (3x – 4) 4. Definition: Binomial Theorem

14 Example: Find the eighth term in the expansion of (x + y) 13. Think of the first term of the expansion as x 13 y 0. The power of y is 1 less than the number of the term in the expansion. The eighth term is 13 C 7 x 6 y 7. Therefore, the eighth term of (x + y) 13 is 1716 x 6 y 7. Example: Find the nth term

15 Homework WS 13-6