Ignoring Fundamental Theorem of Polynomials (a^n + b^n)

[Hotsuma]

Homework Statement

Okay, basically why does (a$$_{}n$$ + b$$_{}n$$) ignore the Fundamental Theory of Polynomials?

Homework Equations

... I could post them here, but basically when n is odd (a$$_{n}$$ + b$$_{n}$$) = a series that looks like this: (a+b) (a$$_{n-1}$$ b$$_{0}$$ - a$$_{n-2}$$b +a$$_{n-3}$$b$$_{2}$$ + ... + a$$_{2}$$b$$_{n-3}$$-a$$_{1}$$b$$_{n-2}$$+b$$_{n-1}$$)

The Attempt at a Solution

I have done a lot on paper, but basically what I am looking for is WHY it isn't following Pascal's triangle. I have flipped through tons of mathematical journals and haven't slept in the past two days for other various reasons... I am doing this for a research class and has lost its proof after years and years of assuming that it is truth.

Argh. If anyone can help I would really appreciate it. If anyone has the proof to this that would be great.

 

[Dick]

I'm not sure what the Fundamental Theorem of Polynomials is, but a^n+b^n factors for odd n but not for even n because a^n=-b^n can be 'solved' for a over the reals for odd n but not for even n. If you work over the complex numbers you can factor the even n case. E.g. a^2+b^2=(a+bi)(a-bi).

 

[HallsofIvy]

Pascal's triangle holds for (x+ y)ⁿ and this isn't of that kind! If you are doing this for a research class perhaps you have something else in mind. My question would be "Why would you expect it to have anything to do with Pascal's triangle?"

 

[D H]

The polynomial $(a^n+b^n)$ doesn't "follow Pascal's triangle" because Pascal's triangle doesn't apply to this polynomial. It applies to the polynomial $(a+b)^n$, which is obviously different from $(a^n+b^n)$ for all n but 1.

 

[Hotsuma]

Okay, fair enough.

I understand that this polynomial does not follow Pascal's triangle and why. That was poorly worded on my part. What I am asking for is the proof for why the factorization of the polynomial exists as it does. I am aware of the fact that it uses imaginary numbers for the even powers. I also know how to derive the (a^n+b^n) formula from the (a^n-b^n) formula. It is complicated and I don't feel like texting the whole thing.

The fundamental theorem of algebra states that every polynomial has n factors based on its degree, yada yada yada. The way the factorization works (If you make the general form) you notice that it establishes a pattern. However, I don't know the steps to factorize it outside of knowing the formula. I am looking to find where this was derived from.

God I wish my brain was being coherent enough to explain what I am trying to say or wasn't being totally lazy and could make up the math text. Sorry.

 

[Hotsuma]

I am about to go on spring break so will be able to work on this more. I am going to get some sleep and then get back to you all.