Binomial Theorem 11.7

Binomial Expansion of the form (a+b)n There are n+1 terms Functions of n Exponent of a in first term Exponent of b in last term Other terms Exponent of a decreases by 1 Exponent of b increases by 1 Sum of exponents in each term is n Coefficients are symmetric (Pascal’sTriangle) At Beginning--increase Towards End---decrease

Expanding Binomials What if the term in a series is not a constant, but a binomial?

Pascal’s Triangle The coefficients form a pattern, usually displayed in a triangle Pascal’s Triangle: binomial expansion used to find the possible number of sequences for a binomial pattern features start and end w/ 1 coeff is the sum of the two coeff above it in the previous row symmetric

Ex 1 Expand using Pascal’s Triangle

Ex 2 Expand using Pascal’s Triangle

Binomial Theorem The coefficients can be written in terms of the previous coefficients

Ex 3 Expand using the binomial theorem

Ex 4 Expand using the binomial theorem

Factorials! factorial: a special product that starts with the indicated value and has consecutive descending factors Ex 5 Evaluate

Binomial Theorem, factorial form and Sigma Notation

Ex 6 Expand using factorial form

Ex 6 Expand using factorial form