CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 12. Discrete Math

12.8 Binomial Theorem

Here you will apply the Binomial Theorem to expand binomials that are raised to a power. In order to do this you
will use your knowledge of sigma notation and combinations.
The Binomial Theorem tells you how to expand a binomial such as( 2 x− 3 )^5 without having to compute the repeated
distribution. What is the expanded version of( 2 x− 3 )^5?

Watch This

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/62265

http://www.youtube.com/watch?v=YxysKtqpbVI James Sousa: The Binomial Theorem Using Combinations

MEDIA

Click image to the left for use the URL below.

URL: http://www.ck12.org/flx/render/embeddedobject/60704

http://www.youtube.com/watch?v=NLQmQGA4a3M James Sousa: The Binominal Theorem Using Pascal’s Trian-
gle

Guidance

The Binomial Theorem states:
(a+b)n=i∑=n 0 (ni)aibn−i

Writing out a few terms of the summation symbol helps you to understand how this theorem works:

(a+b)n=(n 0 )an+(n 1 )an−^1 b^1 +(n 2 )an−^2 b^2 +···+(nn)bn

Going from one term to the next in the expansion, you should notice that the exponents ofadecrease while the
exponents ofbincrease. You should also notice that the coefficients of each term are combinations. Recall that(n 0 )is
the number of ways to choose 0 objects from a set ofnobjects.
Another way to think about the coefficients in the Binomial Theorem is that they are the numbers from Pascal’s
Triangle. Look at the expansions of(a+b)nbelow and notice how the coefficients of the terms are the numbers in
Pascal’s Triangle.