2.4 The Binomial Theorem

Ex 1 Expand and Simplify

(a) (b) (c) (d)

(e) (f) (g)

The expansion of is called the binomial theorem.

A connection that can be made here is that Pascal’s Triangle and Combinations actually work completely in sync with each other.

					1
				1		1
			1		2		1
		1		3		3		1
	1		4		6		4		1
1		5		10		10		5		1

In the case of , the expansion could be interpreted to look like:

With respect to b, we can either choose 0 b’s, 1 b, 2 b’s, or 3 b’s. For any term in this expansion, there is ways of choosing a b, where .

(The same rationale holds for picking a’s, but the for the formula we’re working towards, we should stick to b’s.)

So the expansion of will look like this now:

In general, the expansion of will look like:

Ironically, the simplified version of the formula is:

Ex 2 Expand using the Binomial Theorem.

(a) (b) (c)

p. 293# 1-9