1. Binomial Forms Expansion of Binomial Expressions

2. 10/7/2013 Binomial Forms 2 Special Forms Difference of Squares (Conjugate Binomials) a 2 – b 2 = ( a – b)( a + b) Difference of n th Powers a n – b n = ( a – b)( a n–1 + a n–2 b + a n–3 b 2 +... + a b n–1 + b n–1 ) Polynomial Functions

3. 10/7/2013 Binomial Forms 3 Special Forms Perfect-Square Trinomials ( a + b) 2 = a 2 + 2 a b + b 2 ( a – b) 2 = a 2 – 2 a b + b 2 Binomial Expansion Polynomial Functions n(n – 1) 2 a n–2 b 2 +... + + bnbn n(n–1)...(2) (n – 1)! a b n–1 ( a + b) n = a n + n a n–1 b +

4. 10/7/2013 Binomial Forms 4 Binomial Theorem Expansion of ( a + b ) n Examples: ( a + b ) 2 = a 2 + 2 ab + b 2 ( a + b ) 3 = a 3 + 3 a 2 b + 3 ab 2 + b 3 ( a + b ) 4 = a 4 + 4 a 3 b + 6 a 2 b 2 + 4 ab 3 + b 4

5. 10/7/2013 Binomial Forms 5 Expansion of ( a + b ) n Examples: Binomial Theorem ( a + b ) 5 = a5 +a5 + 1 5 a4ba4b + 5 4 2 1 a3b2a3b2 + 5 4 3 3 2 1 a2b3a2b3 + 5 4 3 2 4 3 2 1 ab 4 + 5! b5b5 = a 5 + 5 a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 ab 4 + b 5 Note: n factorial, n!, is defined as n! = n(n – 1)(n – 2)(n – 3) 3 2 1 and 0! ≡ 1 by definition

6. 10/7/2013 Binomial Forms 6 Expansion of ( a + b ) n The General Case: A Binomial Theorem … + + bnbn k=0 n = ∑ (n–k)! n! k! a n–k b k (a + b)n =(a + b)n = a n + n a n–1 b a n–2 b 2 n(n – 1) 2 + Note: k=0 n ∑ akak = a 0 + a 1 + a 2 + + a n

7. 10/7/2013 Binomial Forms 7 Expansion of ( a + b ) n Pascal’s Triangle Coefficients for ( a + b ) n Binomial Theorem 1 1 1 2 1 1 3 3 1 1 4 6 4 1.... 1 5 10 10 5 1 n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 1 1 2 3 5 8 Fibonacci Sequence Degree

8. 10/7/2013 Binomial Forms 8 1 5 10 10 5 1 1 4 6 4 1 1 3 3 1 1 2 1 1 1 Expansion of ( a + b ) n Pascal’s Triangle Examples: ( a + b ) 0 = Binomial Theorem 1 1 1 2 1 1 3 3 1 1 4 6 4 1.... 1 5 10 10 5 1 1  ( a + b ) 1 = 1 a + 1 b  ( a + b ) 2 =  ( a + b ) 3 =  ( a + b ) 4 =  ( a + b ) 5 1 a 2 + 2 ab + 1 b 2 1 a 3 + 3 a 2 b + 3 ab 2 + 1 b 3 1 a 4 + 4 a 3 b + 6 a 2 b 2 + 4 ab 3 + 1 b 4 = 1 a 5 + 5 a 4 b + 10 a 3 b 2 + 10 a 2 b 3 + 5 ab 4 + 1 b 5

9. 10/7/2013 Binomial Forms 9 Think about it !