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I just ran into this:

$\begin{align} x^1+y^1+z^1&=0\\ x^3+y^3+z^3&=3\\ x^4+y^4+z^4&=15\\ x^2+y^2+z^2&=\text{?} \end{align}$

I know the answer, but can this be solved without guessing?

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    If you ca$n$ afford the CPU cycles, there's [Grobner basis](http://en.wikipedia.org/wiki/Gr%C3%B6bner_basis) :)2012-03-29

2 Answers 2

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Yes, this can be solved without guessing, using Newton's identities.

Since $x + y + z = 0$, they are the roots of $t^3 + at -b = 0$.

Newton's identities give us (in a straightforward mechanical manner) that

$x^3 + y^3 + z^3 = 3b$ $x^4 + y^4 + z^4 = 2a^2$

and $x^2 + y^2 + z^2 = -2a$

This gives us $b=1$ and $2a^2 = 15$.

You can solve for $a$ and find the value of $x^2 + y^2 + z^2 = -2a$. Note that if you assume $x,y,z$ are real, then you need to pick $a \lt 0$ which gives us

$x^2 + y^2 + z^2 = \sqrt{30}$

Note that you did not even need to use the value of $x^3 + y^3 + z^3$.

You can also derive the above identities in a slightly tedious way, as in my answer here: Simplifying an expression $\frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)}$ if we know $x+y+z=0$

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    @MarkBennet: You don't have to apologize! That sentence isn't clear on what the different way is. Your comment helps clarify it a bit. In fact, I was the one being lazy...2012-03-29
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Note that $0 = (x+y+z)^2 = (x^2+y^2+z^2) + 2(xy+xz+yz)$, so $x^2+y^2+z^2 = -2(xy+xz+yz)$.

And $\begin{align*} 15 &= x^4+y^4+z^4\\ & = (x^2+y^2+z^2)^2 - 2(x^2y^2 + x^2z^2 + y^2z^2).\end{align*}$

And $\begin{align*} (xy+xz+yz)^2 &= (x^2y^2 + x^2z^2 + y^2z^2) + 2(x^2yz+xy^2z+xyz^2)\\ &= (x^2y^2+x^2z^2 + y^2z^2) + 2xyz(x+y+z)\\ &= x^2y^2+x^2z^2+y^2z^2 + 2xyz(0)\\ &= x^2y^2+x^2z^2+y^2z^2 \end{align*}$ hence $x^2y^2+x^2z^2+y^2z^2 = (xy+xz+yz)^2.$ So $\begin{align*} 15 &= x^4+y^4+z^4\\ &= (x^2+y^2+z^2)^2 - 2(x^2y^2 + x^2z^2+y^2z^2)\\ &= (x^2+y^2+z^2)^2 - 2(xy+xz+yz)^2\\ &= (x^2+y^2+z^2)^2 -2(xy+xz+yz)(xy+xz+yz)\\ &= (x^2+y^2+z^2)^2 +(x^2+y^2+z^2)(xy+xz+yz)\\ &= (x^2+y^2+z^2)^2 + (x^2+y^2+z^2)\left(-\frac{1}{2}(x^2+y^2+z^2)\right)\\ &= (x^2+y^2+z^2)^2 -\frac{1}{2}(x^2+y^2+z^2)^2\\ &=\frac{1}{2}(x^2+y^2+z^2)^2 \end{align*}$ so $(x^2+y^2+z^2)^2 = 30$, hence $x^2+y^2+z^2=\sqrt{30}$.

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    I'll +1 this answer for just the last comment. (You may know why, by now.) Amazing clarity as ever.+1 :-)2012-03-29