X^n +y^n

I don't see anything that can be expanded.

 

Try alternating signs, and it becomes straightforward.

 

I think you need to think about zeros
[tex]x^n+y^n=0[/tex]
[tex]x^n=-y^n[/tex]
[tex]x=y\cdot\exp(i\pi k/n)[/tex]
[tex]\therefore x^n+y^n=\prod_k (x-\exp(i\pi k/n)y)[/tex]

Occationally combining a subset of these factors together will give you a real solution.

Now you need to think when... :)

 

Then you mean what are the factors

If n is odd, you can factor it as so:

[tex]x^n+y^n=(x+y)(x^{n-1}-x^{n-2}y+x^{n-3}y^2-x^{n-4}y^3+...-xy^{n-2}+y^{n-1})[/tex]

However, if n is even, then [tex]x^n+y^n\neq 0[/tex] except for in the trivial case of x,y=0. This means you can't factor it over the reals. You'll need to use complex numbers. You could convert it into a few different ways, such as [tex]x^n-i^2y^n[/tex] and take difference of two squares, or, if you want to follow the same factorizing process as above, take [tex]x^n+(iy)^n[/tex] and take two cases, when [itexi^n[/itex] is equal to 1, and when equal to -1.