Question

Simplify :

(x-y) ³ + (y-z) ³ + (z-x) ³

Answer 1

( When a+b+c = 0 we know that a³ + b³ + c³ = 3 a b c === Formula *Note it* )

Here lets first take x-y = a
y-z=b and z-x=c then a+b+c=0 So we can apply the above formula

(x-y)³ + (y-z)³ + (z - x)³
= 3 (x - y) (y -z) (z- x)

Or by further simplification
​​​​​​​
= 3 (xy - x z -y² + yz) (z -x)
= 3 (xyz - x² y - x z² + x² z - y² z + x y² + y z² - xyz)
= 3 [ x² (z - y) + z² (y - x) + y² (x - z) ]