Notes 9.5 ~ The Binomial Theorem!

Quick Notes:

Ex. 1: (x + y)6

Expand each of the following.

Ex. 2: (3x – 2y)5

Ex. 3: (x2 + 2y)4

Binomial Theorem�

As long as the exponents stay pretty small, we could use Pascal’s Triangle for each problem

to help us find our coefficients.

However, wouldn’t we like an alternative so that we don’t have to write this out each time we do a problem, especially as the

exponents get larger in value?!

That’s where the Binomial Theorem comes in!

Before we can use the Binomial Theorem, we will need to know how to use the notation for “combinations.”

4C3 is the notation for a situation in which there are 4

things total, taken in groups (combinations) of 3 at a time. You are finding how many different ways you could choose 3 of the 4 things at a time.

Thankfully, there is a formula that allows us to determine the number of ways without us having to think through it each time!

Ex. 4: 5C2 Ex. 5: 8C5 Ex. 6: 8C3

Hmmm…notice

anything about these

last two???

Expand each of the following.

Ex. 7: (3x2 – 2y)36

Ex. 8: (w + z)5

Ex. 9: (2x + y)6

Try these�

for the (r + 1)th term

Ex. 10: Find the 5th term in the

expansion of (3x + 4y)9.

Ex. 11: Find the coefficient of the 3rd

term of the expansion of (4x – y)12.

Did we meet the target?

1. Evaluate: 12C4

2. Find the 3rd term in the expansion of (2x + y)8.