Synopses of Topics - Expanding Binomials

Expanding Binomials

Before getting into the main topic of this section, we must first define some more notation convention.

If n in N and a in R then aⁿ is a real number which results from multiplying n copies of a together. Examples. We define a⁰ = 1 for all a not equal to 0 and 0⁰ as undefined.

When these terms are multiplied together, there are a few general rules to make the calculations simplier.
If a, b in R and n, m in N then

aⁿa^m = a^n+m
(aⁿ)^m = a^nm
(ab)ⁿ = aⁿbⁿ

Check here for examples.

Just as you can not forget about brackets when addition and multiplication are mixed
(i.e. a(b+c) =ab + ac), the same is true for when you have addition raised to some power. Here are some examples of what we are talking about.

From the examples above, you probably noticed a pattern in the expansions. If (a+b)ⁿ is expanded out, it would be the sum of n+1 terms and each term is some natural number multiplied by a^n-jb^j where j = 0, 1, 2, ..., n successively. The natural number is given by Cⁿ_j , nCj or nCj , where all of these are read as "n choose j".

Try doing the examples above in this type of notation. If you get stuck, just click here.
We then get the general formula (where a, b in R and n in N): (a+b)^n = nC0*a^n + nC1*a^(n-1)b + nC2*a^(n-2)b^2 +...+ nCn*b^n
This is the BINOMIAL EXPANSION.

But how do we calculate nCj ? There are two different ways. The first one is using Pascal's triangle. What's that you ask? Well it is best to show you a picture. As you might well imagine, this would get very tiresome if n was large. Therefore we use the second method - which involves the use of factorial notation. All that means is taking a number, say k, and multiplying together the numbers from 1 to k. It is written as k! (read as k factorial) and we say k! = 1*2*3*...*k. We define 1! = 1 and 0! = 1. Note that k! = k(k-1)! Examples. Using this, we define
nCj = n! / (j!(n-j)!)

Now we can prove that the formula nCj = (n-1)C(j-1) + (n-1)Cj used for creating Pascal's Triangle does, in fact, hold true. Try doing it yourself frist before looking at the proof.

Here are some examples of the binomial expansion using what we have just learned.