## Fun with the distributive property

$(x+1)(x-1) = (x^2) − 1$

And in general,

$(a+b)(a-b) = (a^2) − (b^2)$

## Polynomials are neat

Any polynomial of odd degree with real coefficients must have at least one real root and an even number of complex roots. (That number can be zero.)

## A cool observation about powers of 2

$2^0 + 2^1 = 2^2 -1$

$2^0 + 2^1 + 2^2 = 2^3 - 1$

In general, $2^0 + 2^1 + \dots + 2^{n-1} = 2^n -1$. But wait, there’s more!

It’s well known that $2^{-1} + 2^{-2} + 2^{-3} + \dots = 1$ — in fact you can represent this in binary as $0.\overline{111}\dots_{2} = 1$ (in much the same way that $0.\overline{999}\dots_{10}=1$).

But *this* means that:

$2^{n-1} + 2^{n-2} + \dots + 2^0 + 2^{-1} + \dots = 2^n$

Which is rather satisfying, is it not?