what is this????
When does x^3+y^3+z^3-3xyz equals to zero?
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1 solution
This doesn't prove there are no other equality cases.
x 3 + y 3 + z 3 − 3 x y z = ( x + y + z ) ( x 2 + y 2 + z 2 − x y − y z − z x ) = 0
⟺ either x + y + z = 0 or
x 2 + y 2 + z 2 − x y − y z − z x = 0
⟺ ( x − y ) 2 + ( y − z ) 2 + ( z − x ) 2 = 0 ⟺ x = y = z .
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This is the standard solution. Nice job.
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William, just key in "\ " and "\ " before and after x^3 + y^3 + z^3 - 3xyz and you will get perfect x 3 + y 3 + z 3 − 3 x y z in LaTex.
x 3 + y 3 + z 3 x 3 + y 3 + z 3 − 3 x y z = ( x + y + z ) ( x 2 + y 2 + z 2 ) − ( x y + y z + z x ) ( x + y + z ) + 3 x y z = ( x + y + z ) ( x 2 + y 2 + z 2 ) − ( x y + y z + z x ) ( x + y + z )
When x + y + z = 0 :
⇒ x 3 + y 3 + z 3 − 3 x y z = ( 0 ) ( x 2 + y 2 + z 2 ) − ( x y + y z + z x ) ( 0 ) = 0
When x = y = z :
⇒ x 3 + y 3 + z 3 − 3 x y z = 3 x 3 − 3 x 3 = 0