Fraction Decomposition I have the following problem:

Suppose $x + y + z = 0$. Show that $$\frac{x^5 + y^5 + z^5}{5}= \frac{x^3 + y^3 + z^3}{3} \times \frac{x^2 + y^2 + z^2}{2}$$ and $$\frac{x^7 + y^7 + z^7}{7}= \frac{x^2 + y^2 + z^2}{2} \times \frac{x^5 + y^5 + z^5}{5}$$

I thought this was a fun problem to tackle, but I've been stuck on this for hours, and I can't seem to get anywhere. I've tried expanding, but no matter what I do, I just get a huge mess.
I also cannot find how to incorporate the initial assumption ($x + y + z = 0$) anywhere. Have I missed something?

The first identity does not involve much algebra, if you write $z = -(x+y)$:
$$ z^5 + (x^5+y^5) = -5x^4y-10x^3y^2-10x^2y^3-5xy^4$$
where the $(-x^5-y^5)$ cancels out in the binomial exapnsion.
A similar thing happends in the cubes term.
So the identity reads
$$
-x^4y-2x^3y^2-2x^2y^3-xy^4 = (-x^2y-xy^2)\frac{(x^2+y^2+x^2+2xy+y^2)}{2}
$$
and of the six terms on the right, two pairs add together, and you get the left side.
