# What is the expansion of $(1 + x)^n$?

Forgive the total beginner question, it's been a few years since I needed to do this stuff.

What is the expansion of $(1 + x)^n$?

-
It's already pretty simplified. I think you want its expansion, and for that you should look up the binomial theorem. – Qiaochu Yuan Nov 1 '10 at 23:50
That's as simple as it can get. What exactly do you need, because it seems you don't need the "simple" expression... – J. M. Nov 1 '10 at 23:50
You could start with n=2 and use the distributive law. Then you will be on the way for n=3 and may see a pattern. – Ross Millikan Nov 1 '10 at 23:52
It will be 1^n + x^n, that is, assuming n is prime and we are in a field with characteristic n ! – crasic Nov 2 '10 at 2:19

$$1 + n$$

$$1 + 2n + n^2$$

$$1 + 3n + 3n^2 + n^3$$

$$1 + 4n + 6n^2 + 4n^3 + n^4$$

$$1 + 5n + 10n^2 + 10n^3 + 5n^4 + n^5$$

$$1 + 6n + 15n^2 + 20n^3 + 15n^4 + 6n^5 + n^6$$

See if you can spot a pattern!

-
Thanks muad. Generalizing the expansion would take the notation used on the binomial theorem page, I presume, with coefficients matching entries in pascals triangle. I will have to work out how to insert mathematical notation here. – Ben Nov 2 '10 at 0:19
In binomial expansion its written as the sum over all the terms, that is $(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k}y^k$ – crasic Nov 2 '10 at 2:24