Why in the binomial expansion are the powers of x positive? In Binomial expansion with $n$ as a rational, positive number then,
$$(x+y)^n =\binom{n}{0}y^n + \binom{n}{1}  y^{n-1}  x +\binom{n}{1}  y^{n-2}x^2 +\cdots+\binom{n}{n}  x^{n}$$
While if $\alpha$ is a rational, non-positive number and $\lvert x \rvert< 1$ then,
$$(1+x)^\alpha = 1 + \binom{\alpha}{1} x + \binom{\alpha}{2}  x^2 +\cdots$$
Imagine if $\alpha$ is $-3$ then how can $x$ be going up in positive powers?

Here is one way to see this - from the theory of GP, it may be familiar to write for a common ratio $|x| < 1$,
$$\frac1{1-x} = 1+x+x^2+x^3 + \cdots$$
Using $x \to -x$, we get 
$$\frac1{1+x} = 1-x+x^2-x^3 + \cdots$$
If you differentiate that, you get
$$-\frac1{(1+x)^2} = -1+2x-3x^2+4x^3 -\cdots$$
$$\implies \frac1{(1+x)^2} = 1-2x+3x^2-4x^3 +\cdots$$
I hope you can extend easily to the case $\alpha = -3$...
