Fraction Decomposition

Question

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?

Answer 1

You've surely been trying to express one of the unknowns using the others and plugging in. Try to preserve the symmetry between them instead:

$$(x+y+z)^3 = x^3 + y^3 + z^3 + 3x^2y + 3x^2z + 3y^2x + 3y^2z + 3z^2x + 3z^2y + 6xyz$$

But we know that the left hand side is a zero and, moreover, wherever something like $x+z$ appears we can replace that by a $-y$ (again, strictly keeping exchange symmetry). So the same equation can be written as

$$0 = x^3 + y^3 + z^3 + 3x^2(-x) + 3y^2(-y) + 3z^2(-z) + 6xyz$$

and even better than that!

$$\begin{aligned} 0 &= -2(x^3 + y^3 + z^3) + 6xyz \\ xyz &= \frac{x^3 + y^3 + z^3}3. \end{aligned}$$

Surely replacing the right hand side by the left in your problem will make the multiplications simpler! Try to proceed from here :-)

Hint: apart from $S_{111} = xyz$ and $S_k = x^k+y^k+z^k$, among which you have a first relation here, you'll need other symmetric forms like $S_{11} = xy+xz+yz$.

Update: I have been able to derive both your equations this way so it's guaranteed to work. It just takes some time. There's a whole theory about monomial and power-sum symmetric polynomials but I didn't want to go into that.

Starting from $S_1 = 0$, you can derive things like $$\begin{aligned} S_2 &= -2S_{11} \\ S_3 &= 3S_{111} \\ S_4 &= \frac12 S_2^2 \end{aligned}$$ and also some useful observations like $$\begin{aligned} S_{kl} &= S_k S_l - S_{k+l} \\ S_{(a+c)(b+c)c} &= S_{ccc} S_{ab} = S_{111}^c S_{ab} \end{aligned}$$ With these you're one little theorem away from $S_5/5 = S_3S_2/6$ and $S_7/7 = S_5S_2/10$.

Answer 2

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.