What is the formula of x^3 - y^3 - z^3 -3xyz ???

1 Answer
Sonnhard
Jul 10, 2018

(x-y-z)(x^2+xy+y^2+xz-yz+z^2)#

Explanation:

Proof:
Note that #x=y+z# is a solution of

#x^3-y^3-z^3-3xyz=0#
plugging #x=y+z# in the equation above:

#(y+z)^3-y^3-z^3-3(y+z)yz=y^3+3y^2z+3yz^2+z^3-y^3-z^3-3y^2z-3yz^2=0#

so we can divide #x^3-y^3-z^3-3xyz# divide by #x-y-z#
and we get

#x^2+xy+y^2+xz-yz+z^2#

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