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

In summary, the homework equation (a^n+b^n) does not factor for odd n, but it does for even n. The fundamental theorem of algebra states that every polynomial has n factors, and this is based on its degree. However, the way the factorization works is that it establishes a pattern.
  • #1
Hotsuma
41
0

Homework Statement



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

Homework Equations



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

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.
 
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  • #2
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).
 
  • #3
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?"
 
  • #4
The polynomial [itex](a^n+b^n)[/itex] doesn't "follow Pascal's triangle" because Pascal's triangle doesn't apply to this polynomial. It applies to the polynomial [itex](a+b)^n[/itex], which is obviously different from [itex](a^n+b^n)[/itex] for all n but 1.
 
  • #5
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.
 
  • #6
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.
 

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

What is the Fundamental Theorem of Polynomials?

The Fundamental Theorem of Polynomials states that every polynomial function of degree n has n complex roots, where some roots may be repeated.

What does it mean to ignore the Fundamental Theorem of Polynomials?

Ignoring the Fundamental Theorem of Polynomials means that we are not considering or following the rules and principles outlined in the theorem.

What happens if we ignore the Fundamental Theorem of Polynomials?

If we ignore the Fundamental Theorem of Polynomials, it means that we are not taking into account the full range of possibilities for polynomial functions and their roots. This could lead to incorrect solutions or a lack of understanding of the behavior of polynomial functions.

Why is it important to follow the Fundamental Theorem of Polynomials?

The Fundamental Theorem of Polynomials is important because it provides a fundamental understanding of the behavior of polynomial functions and their roots. It allows us to accurately solve and analyze polynomial equations and functions.

What are some potential consequences of ignoring the Fundamental Theorem of Polynomials?

Ignoring the Fundamental Theorem of Polynomials can lead to incorrect solutions, misunderstandings of polynomial functions, and a lack of understanding of the relationship between polynomial functions and their roots. It can also hinder the ability to solve more complex polynomial equations and problems.

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