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I want to have a decomposition of this :

$$(x^2+y^2+z^2)(x+y+z)(x+y-z)(-x+y+z)(x-y+z)-8x^2y^2z^2$$

I have tried all possible calculation which came to my mind,I will describe one of it which is better but no result:

I put $(x+y-z)=a$ ,$ (-x+y+z)=b$ and $(x-y+z)=c$ and we have $(x+y+z)=a+b+c$ so we have $$(x^2+y^2+z^2)(a+b+c)abc-8x^2y^2z^2$$

but I couldn't make it simpler, it will be great if you help me about it,thanks.

kpax
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    the product of the four linear terms is well known, it comes from Heron's formula for the area of a triangle but is multiplied by $16$ or the like. http://en.wikipedia.org/wiki/Heron%27s_formula – Will Jagy May 21 '15 at 20:32
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    Most useful version for this seems to be $ (x+y+z)(x+y-z)(-x+y+z)(x-y+z) = $ $(x^2 + y^2 + z^2)^2 - 2(x^4 + y^4 + z^4)$ – Will Jagy May 21 '15 at 20:36
  • actually I have tried it, but I will check it again. – kpax May 21 '15 at 20:39

1 Answers1

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Maple does the job by

factor(expand((x^2+y^2+z^2)*(x+y+z)*(x+y-z)*(-x+y+z)*(x-y+z)-8*x^2*y^2*z^2));

$$ - \left( {x}^{2}-{y}^{2}-{z}^{2} \right) \left( {x}^{2}+{y}^{2}-{z}^{ 2} \right) \left( {x}^{2}-{y}^{2}+{z}^{2} \right) .$$

user64494
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  • We humans prefer the more symmetric $\left( {x}^{2}+{y}^{2}-{z}^{ 2} \right) \left( {y}^{2}+{z}^{2}-{x}^{ 2} \right) \left( {z}^{2}+{x}^{2}-{y}^{ 2} \right) $ – Georges Elencwajg May 21 '15 at 20:56
  • @ Georges Elencwaig : The Maple developers are not foolish. – user64494 May 21 '15 at 20:58
  • it seems that doing it without computer software is not an easy job! thanks a lot. – kpax May 21 '15 at 21:03
  • @ Georges Elencwaig : I think some general algorithm is applied which produces minus as side effect. – user64494 May 21 '15 at 21:11
  • Yes, that's right. The point is that results given by software are indeed practically always correct but are sometimes given in a form that is not what a mathematician expects, due to the use of general powerful algorithms . So it may sometimes be advantageous to slightly modify the presentation of the result. – Georges Elencwajg May 21 '15 at 21:19