PreCalculus B 5th Hour Winter 11: Section 9.5: The Binomial Theorem

A binomial is a polynomial that has two terms. This section will teach you how to raise a binomial to a power in a relatively quick way.

The Binomial Theorem:

nCr=

For example, you have the following toppings at a pizza store:

Pepperoni

Sausage

Onion

Ham

Bacon

You are allowed to put 2 toppings on each pizza. How many different combinations of toppings do you have?

n=5

r=2

There are 77,520 different combinations of pizza

Pascal's Triangle

Pascal's Triangle starts with row 0 and gradually gets larger.

Pascal's Triangle contains many different patterns, but we are going to focus on how the numbers in each row are similar to the coefficients when raising (x+y) to different powers.

(x+y)^0=1

(x+y)^1=1x+1y

(x+y)^2=1x^2+2xy+1y^2

(x+y)^3=1x^3+3x^2y+3xy^2+1y^3

(x+y)^4=1x^4+4x^3y+6x^2y^2+4xy^3+1y^4

(x+y)^5=1x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+1y^5

The red numbers are the coefficients and make the shape and pattern of Pascal's Triangle.

Lets try an example:

1x^5+5x^4*4+10x^3*4^2+10x^2*4^3+5x*4^4+1*4^5

*Note: The binomial is raised to the 5th power. This tells you to look at row 5 of Pascal's Triangle and the numbers in this row will be the coefficients