sequences and series - Repeatedly taking differences on a polynomial yields the factorial of its degree?

Consider a function such that it takes in polynomial function and creates an array of its outputs and then using that array creates another new array by calculating the absolute difference between the first $2$ values and keeps doing this until it reaches an array full of zeros.

This is much easier to show you by example.

For example take $F(x)= x^2$, the first array would be

$1,4,9,16,25,36,49,64,81$ and so on, the second would be

$3,5,7,9,11,13,15,17,19$ ( the difference between the first value and the second one)

but the third one is where it gets interesting as if we continue the pattern we would get an array filled with only $2$'s and after that it would only be zeros.

Lets do another example, $F(x)=x^3$

$1,8,27,64,125,216,343$

$7,19,37,61,91,\dotsc$ but here is the interesting part if we continue this

$12,18,24,30,\dotsc$ and once more then we get

$6,6,6,6,6,\dotsc$ after that it would just be an array of zeros

There are $2$ main observation that I made about this

Firstly, the value that is begin repeated indefinitely is equals to the factorial of the functions power. Meaning that for $F(x)=x^2$ the value being repeated is $2!$. For $F(x)=x^3$ , it's $3! $ and this is true for all polynomials (I tried it up to $x^7$, after that it got too messy)

Secondly, the value that is repeated always occurs on the $n$th iteration of the function. Meaning that for $F(x)=x^2$, we have to go through the processes $2$ times before we find the value. For $F(x)=x^3$, we have to go through it $3$ times before getting the value.

Is there any way to prove this and does this mean anything at all?