Chapter 8 Sec 5 The Binomial Theorem

2 of 15 Pre Calculus Ch 8.5 Essential Question How do you find the expansion of the binomial (x + y) n ? Key Vocabulary: Binomial Theorem Pascal’s Triangle

3 of 15 Pre Calculus Ch 8.5 Binomial Coefficients To begin this section, lets look at the expansion of (x + y) n for several values of 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.In each there are n + 1 terms 2.In each x and y have symmetrical roles, power of x decrease by 1 and y increases by 1. 3.Sum of the powers equals n. 4. The coefficients increase then decrease symmetrically.

4 of 15 Pre Calculus Ch 8.5 Binomial Theorem

5 of 15 Pre Calculus Ch 8.5 Example 2 Find each binomial coefficient a. 7 C 3 b. 7 C 4 c. 12 C 1 d. 12 C 11

6 of 15 Pre Calculus Ch 8.5 Example 3 Write the expansion of the expression (x + 1) 3. The binomial coefficients are 3 C 0 = 1, 3 C 1 = 3, 3 C 2 = 3, 3 C 3 = 1 1.In each there are n + 1 terms 2.In each x and y have symmetrical roles, power of x decrease by 1 and y increases by 1. 3.Sum of the powers equals n. 4.The coefficients increase then decrease symmetrically.

7 of 15 Pre Calculus Ch 8.5 Example 4 Write the expansion of the expression (x – 1) 3. The binomial coefficients are 3 C 0 = 1, 3 C 1 = 3, 3 C 2 = 3, 3 C 3 = 1

8 of 15 Pre Calculus Ch 8.5 Example 5 Write the expansion of the expression a. (2x – 3) 4 b. (x – 2y) 4 The binomial coefficients are 4 C 0 = 1, 4 C 1 = 4, 4 C 2 = 6, 4 C 3 = 4, 4 C 4 = 1

9 of 15 Pre Calculus Ch 8.5 Example 6 Write the expansion of the expression (x 2 + 4) 3 The binomial coefficients are still 3 C 0 = 1, 3 C 1 = 3, 3 C 2 = 3, 3 C 3 = 1 3 C 0 = 1, 3 C 1 = 3, 3 C 2 = 3, 3 C 3 = 1 (x 2 + 4) 3 = (1)(x 2 ) 3 + (3)(x 2 ) 2 (4) + (3)(x 2 )(4) 2 + (1)(4) 3 = x x x

10 of 15 Pre Calculus Ch 8.5 Example 7 Sometimes you will need to find a specific term in a binomial expansion. From the Binomial Theorem the (r + 1) th term is n C r x n – r y r a. Find the sixth term of (a + 2b) 8. a. Find the sixth term of (a + 2b) 8. To find the sixth term, use n = 8 and r = 5 {the formula is for the (r + 1) st term, so r is one less than the number of the term you are looking for} a. 8 C 5 a 8 – 5 (2b) 5 = 56 ∙ a 3 ∙ (2b) 5 = 56(2 5 )a 3 b 5 = 1792a 3 b 5 a. 8 C 5 a 8 – 5 (2b) 5 = 56 ∙ a 3 ∙ (2b) 5 = 56(2 5 )a 3 b 5 = 1792a 3 b 5 b. Find the coefficient of the term a 6 b 5 in the expansion of (2a – 5b) 11. b. Find the coefficient of the term a 6 b 5 in the expansion of (2a – 5b) 11. b. 11 C 5 (2a) 6 (–5b) 5 = (462)(64a 6 )(–3125b 5 ) = – 92,400,000a 6 b 5 n = 11, r = 5, x = 2a, y = –5b

11 of 15 Pre Calculus Ch 8.5 Pascal’s Triangle Pascal gave us a convenient way to remember the pattern for binomial coefficients First and last numbers are

12 of 15 Pre Calculus Ch 8.5Pascal’s Lets put them together x + 1 y 1 x 2 + 2xy + 1 y 2 1 x 3 + 3x 2 y + 3xy y 3 1 x 4 + 4x 3 y + 6x 2 y 2 + 4xy y 4 1 x 5 + 5x 4 y + 10x 3 y x 2 y 3 + 5xy y 5 1 x 6 + 6x 5 y + 15x 4 y x 3 y x 2 y 4 + 6xy y 6 1 x 7 + 7x 6 y + 21x 5 y x 4 y x 3 y x 2 y 5 + 7xy y 7 (x + y) 0 = (x + y) 1 = (x + y) 2 = (x + y) 3 = (x + y) 4 = (x + y) 5 = )6 =)6 =)6 =)6 =

13 of 15 Pre Calculus Ch 8.5 Example 8 Use the seventh row of Pascal’s Triangle to find the binomial coefficients C 0, 8 C 1, 8 C 2, 8 C 3, 8 C 4, 8 C 5, 8 C 6, 8 C 7, 8 C 8 Write the seventh row of Pascal’s Triangle C 0 8 C 1 8 C 2 8 C 3 8 C 4 8 C 5 8 C 6 8 C 7 8 C 8

14 of 15 Pre Calculus Ch 8.5 Essential Question How do you find the expansion of the binomial (x + y) n ?

15 of 15 Pre Calculus Ch 8.5 Daily Assignment Chapter 8 Section 5 Text Book Pg 624 – 625 # 1 – 33 Mode 4, 49 – 53 Odd; 57, 61, 69, 71, 79, 85 Read Section 8.6 Show all work for credit.