X^n +y^n

[phymatter]

Main Question or Discussion Point

What is the expansion of xⁿ +yⁿ , when is even ??/

 

[elibj123]

I don't see anything that can be expanded.

 

[phymatter]

I don't see anything that can be expanded.

i mean that xⁿ - yⁿ can be written as (x-y)(x^n-1 +x^n-2y ....+y^n-1 )
similarly what can xⁿ +yⁿ be written as ????????

 

[jav]

Try alternating signs, and it becomes straightforward.

 

[Gerenuk]

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

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

Now you need to think when... :)

 

[Mentallic]

What is the expansion of xⁿ +yⁿ , when is even ??/

i mean that xⁿ - yⁿ can be written as (x-y)(x^n-1 +x^n-2y ....+y^n-1 )
similarly what can xⁿ +yⁿ be written as ????????

Then you mean what are the factors

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

$$x^n+y^n=(x+y)(x^{n-1}-x^{n-2}y+x^{n-3}y^2-x^{n-4}y^3+...-xy^{n-2}+y^{n-1})$$

However, if n is even, then $$x^n+y^n\neq 0$$ 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 $$x^n-i^2y^n$$ and take difference of two squares, or, if you want to follow the same factorizing process as above, take $$x^n+(iy)^n$$ and take two cases, when [itexi^n[/itex] is equal to 1, and when equal to -1.