Binomial Theorem

What happens when you multiply a binomial by itself ... many times?

Here is the answer:

Don't worry ... I will explain it!

Binomial

A binomial is a polynomial with two terms

example of a binomial

Multiplying

The Binomial Theorem shows what happens when you multiply a binomial by itself (as many times as you want).

It works because there is a pattern ... let us see if we can discover it.

Exponents

But first you need to know what an Exponent is. Here is a quick summary: An exponent says how many times to use something in a multiplication. In this example: 82 = 8 × 8 = 64

An exponent of 1 means just to have it appear once, so you get the original value:

An exponent of 0 means not to use it at all, and so we have only 1:

Exponents of (a+b)

Now on to the binomial.

We will use the simple binomial a+b, but it could be any binomial.

Let us start with an exponent of 0 and build upwards.

Exponent of 0

When an exponent is 0, you get 1:

(a+b)⁰ = 1

Exponent of 1

When the exponent is 1, you get the original value, unchanged:

(a+b)¹ = a+b

Exponent of 2

An exponent of 2 means to multiply by itself (see how to multiply polynomials):

(a+b)² = (a+b)(a+b) = a² + 2ab + b²

Exponent of 3

For an exponent of 3 just multiply again:

(a+b)³ = (a+b)(a² + 2ab + b²) = a³ + 3a²b + 3ab² + b³

We have enough now to start talking about the pattern.

The Pattern

In the last result we got:

a³ + 3a²b + 3ab² + b³

Now, notice the exponents of a. They start at 3 and go down: 3, 2, 1, 0:

Likewise the exponents of b go upwards: 0, 1, 2, 3:

If we number the terms 0 to n, we get this:

Which can be brought together into this:

a^n-kb^k

It works like magic!

Coefficients

We are missing the numbers (which are called coefficients).

Let's look at all the results we got before, from (a+b)⁰ up to (a+b)³:

And now look at just the coefficients (with a "1" where a coefficient wasn't shown):

They actually make Pascal's Triangle! Each number is just the two numbers above it added together (except for the edges, which are all "1") (Here I have highlighted that 1+3 = 4)

Armed with this information let us try something new ... an exponent of 4:

a exponents go 4,3,2,1,0:		a4	+	a3	+	a2	+	a	+	1
b exponents go 4,3,2,1,0:		a4	+	a3b	+	a2b2	+	ab3	+	b4
coefficients go 1,4,6,4,1:		a4	+	4a3b	+	6a2b2	+	4ab3	+	b4

We have success!

We can now use that pattern for exponents of 5, 6, 7, ... 50, ... 112, ... you name it!

As a Formula

Our last step is to write it all as a formula.

But hang on, how do we write a formula for "find the coefficient from Pascal's Triangle".

Well, there is such a formula:

It is commonly called "n choose k" because it is how many ways to choose k elements from a set of n. The "!" means "factorial", for example 4! = 1×2×3×4 = 24

Putting It All Together

The last step is to put all the terms together into one formula.

But we are adding lots of terms together ... can that be done using one formula?

Yes! The handy Sigma Notation allows us to sum up as many terms as we want:

Sigma Notation

Now it can all be put together in one formula:

Use It

Let us now use it for n = 3 :

And there you have it.

Isaac Newton

As a footnote it is worth mentioning that around 1665 Sir Isaac Newton came up with a "general" version of the formula that is not limited to exponents of 0, 1, 2, .... I hope to write about that one day.