Victoria School AM #6 Binomial Theorem 2024

Secondary 4 Additional Mathematics Binomial Theorem 1 For a positive integer n, ! ( 1) ( 2) ... 3 2 1n n n n=  −  −     Teacher Name: Class Reg No. Date: 4 ___ Chapter 6: Binomial Theorem Reference Book: Additional Maths 360 Textbook, Marshall Cavendish You will learn how to, • Use the notations !n and ,n r   • Expand (1 + b) n and (a + b) n to any number of terms, where n is a positive integer, • Identify and find a particular term in the expansion of (1 + b) n using the result Tr + 1 = rn      b r , • Use the result Tr + 1 = rn      a n – r b r to find a particular term in the expansion of (a + b) n , • Use the general term formula to find the specific terms, coefficients and unknown values. The factorial n! The notation n! is read as “ n factorial”. It is defined for a positive integer n as the product of the first n positive integers. In mathematical terms, Recap… Recall that ( ) 2 22 2a b a ab b+ = + + . Hence, when 1a = , we have ( ) 2 21 1 2b b b+ = + + . In pairs, expand the following and arrange the terms in ascending powers of b. (a) ( ) 3 1 b+ ( )( ) ( )( ) 2 2 2 2 3 23 1 1 1 1 2 1 2 2 1 3 3 b b b b b b b b b b b b b + + = + + + = + + + + + = + + + (b) ( ) 4 1 b+ ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( ) 4 2 2 22 43 23 43 23 2 3 2 3 2 3 2 3 4 2 3 4 1 1 1 1 2 1 2 or 1 1 1 1 1 3 3 1 1 1 1 1 3 3 1 1 3 3 1 3 3 1 3 3 3 3 1 4 6 4 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b + = + + = + + + + + = + + = + + + + + = + + = + + + + = + + + + + + + = + + + + + + + = + + + +

Secondary 4 Additional Mathematics Binomial Theorem 2 (c) ( ) 5 1 b+ ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) 5 2 3 2 2 3 2 3 2 3 2 2 3 2 3 2 3 4 2 3 4 5 2 3 4 5 1 1 1 1 2 1 3 3 1 1 3 3 2 1 3 3 1 3 3 1 3 3 2 6 6 2 3 3 1 5 10 10 5 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b + = + + = + + + + + = + + + + + + + + + + + = + + + + + + + + + + + = + + + + + (d) ( ) 6 1 b+ ( ) ( )( ) ( )( ) ( ) ( ) 65 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 6 2 3 4 5 6 1 1 1 1 1 5 10 10 5 1 1 5 10 10 5 1 5 10 10 5 1 5 10 10 5 5 10 10 5 1 6 15 20 15 6 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b + = + + = + + + + + + = + + + + + + + + + + + = + + + + + + + + + + + = + + + + + + Is there a faster way to expand ( )1 n b+ for all positive integer n?

Secondary 4 Additional Mathematics Binomial Theorem 3 Pascal Triangle Complete the following array by adding up the adjacent numbers in the preceding row above. This is known as the Pascal Triangle, named after the mathematician, Blaise Pascal, though the ancient Chinese were known to use it before he was born. One of its many interesting properties is that each row in the array corresponds to the coefficients in a binomial expansion. These coefficients are also