We will first prove the distributive property:
QED
Now, for
Let .
QED

We will first prove the distributive property:
QED
Now, for
Let .
QED
It is not clear what you have assumed as your starting point or what you are really trying to do. But this basilcally gets you the distributive property for natural numbers, assuming that you already have the commutative and associative properties for addition. It does not get you the distributive property in any larger context.Originally Posted by Ellatha
This is just repeated applications of the distributive property, where the distributive property is already known to hold. It has nothing particulatly to do with polynomials.Originally Posted by Ellatha
What are you trying to do. and in what specific context are you trying to do it ? (i.e. are looking at ordinary real polynomials, the polynomial ring over a commutative ring, or what ?)
If you want to see arithmetic developed from the Peano Postulates, take a look at Landau's Foundations of Analysis. Warning  it is very dry.
Well, at the moment I'd just like to write a few proofs regarding various theorems and properties in mathematics. In all of these are proofs I haven't used any resources, so they may very well be subject to error. In the second case I was merely showing that the product of polynomials can be derived via treating one polynomial as a single variable than distributing, followed by substitution. Here I'm considering natural numbers and univariate polynomials over a ring R. I think this is a much better way of looking at polynomial multiplication as opposed to "FOILing" or other such computationbased algorithms.Originally Posted by DrRocket
Does the second proof also prove the binomial theorem? It would seem to do so (although the method described would be more tedious than that developed by Newton or Pascal).Originally Posted by DrRocket
noOriginally Posted by Ellatha
It doesn't prove anything of note. As I said it is just repeated applications of the distributive property.
What distinguishes the binomial theorem from a repeated application of the distributive property? I do know that the application I provided could find any (a + b)^n for all n as an element of the natural numbers.
The binomial theoren yields a closed form experession for (a+b)^n as a polynomial in a and b, not just an algorithm for expanding it to eventually find one.Originally Posted by Ellatha
I totally agree with you that polynomials have some bad properties! However, we need them for different research applications, e.g. polynomial chaos expansion (see a recent paper on that here http://www.oladyshkin.net/publicatio...in_al_2009.pdf ) etc.
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