9.5 The Binomial Theorem

A binomial is a polynomial that has two terms.

Lets look at the expansion of

n = 0

n=1

n=2

n=3 ... etc.

Observations:

1. In each expansion, there are n+1 terms

2. In each expansion, x and y have symmetric roles. The powers of x decrease by one in successive terms, whereas the powers of y increase by one

3. The sum of the powers in each term in n. For instance, in the expansion of (x+y)^5, the sum of the powers in each term is 5.

4. The coefficients increase then decrease in a symmetric pattern.

The coefficients of a binomial expansion are called Binomial Coefficients. To find them, you can use the Binomial Theorem.

The Binomial Theorem:

In the expansion of

the coefficient of is:

Pascal's Triangle:

By arranging the coefficients in a triangular pattern, you obtain the following array, which is called Pascal's Triangle.

->The first and last number in each row of Pascal's Triangle is 1. Every other number in each row is formed by adding the two numbers immediately above the number. Pascal noticed that numbers in this triangle are precisely the same numbers as the coefficients of binomial expansions.

->The top row in Pascal's Triangle is called the zero row because it corresponds to the binomial expansion

->The nth row in Pascal's Triangle gives the coefficients of

Binomial Expansions:

When you write out the coefficients for a binomial that is raised to a power, you are expanding a binomial.