Long ago, when we studied polynomials, we learned that a binomial is an expression that has two terms. In this lesson, we are going to learn how to expand a binomial for any arbitrary power n. If we have any integer n, we will be able to use this to figure out how to expand (a + b) raised to the n. You'll learn the formula and you'll use the factorials in it. Besides that, you'll learn how to find the binomial coefficients with Pascal's triangle. This lesson has a proof of the binomial theorem at the end which can be interesting to see, but you don't have to.
The binomial theorem tells us what happens when we raise a+b to some arbitrary power n: (a+b)n.
Notice that when we expand (a+b)n, it will always come out in the form
an + an−1b + an−2b2 + …+ a2 bn−2 + a bn−1 + bn .
We call each of the blanks a binomial coefficient since they are the coefficients of the binomial expansion.
Binomial theorem: When expanding (a+b)n, the coefficient of the term an−kbk is
[Remember, (n || k) is spoken as `n choose k'. We can also write it as nCk or C(n,k). We first learned about it in the lesson Permutations and Combinations. Also remember, `!' means `factorial': the value we get when we multiply
a number by all the positive integers below it: 5! = 5·4 ·3 ·2 ·1. We define 0! = 1 for ease and other fiddly reasons.]
We can also find the binomial coefficients with Pascal's triangle. We start with a 1 at the very top of the triangle for n=0, then each row below is created by diagonal addition. [This idea is hard to explain with words but makes a lot more sense
in pictures. Check out the video for a diagram of what's going on and how to use it.]
If you're interested, this lesson has a proof of the binomial theorem at the end. Don't worry about watching it if you don't want to, but if you do, great! Working through proofs like this is good experience if you're interested in eventually studying
The Binomial Theorem
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.