11.711.7

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

na b

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

0a b

1a b

2a b

4a b

3a b

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 1Expand using Pascal’s Triangle 5p q

Ex 2Expand using Pascal’s Triangle 6x y

The coefficients can be written in terms of the previous coefficients

0 1 1 2 2 3 3 01 1 21 ... 1

1 1 2 1 2 3

n n n n n nn n n n nna b a b a b a b a b a b

Ex 3Expand using the binomial theorem

8x y

Ex 4Expand using the binomial theorem

45x

factorial: a special product that starts with the indicated value and has consecutive descending factors

Ex 5Evaluate6!

2!4!

0 1 1 2 2 0! ! ! !

...!0! 1 !1! 2 !2! 0! !

n n n n nn n n na b a b a b a b a b

n n n n

or

0

!

! !

nn n k k

k

na b a b

n k k

Ex 6Expand using factorial form 43x y

Ex 6Expand using factorial form 42 3x