Show that
$$(x-y)^3+(y-z)^3+(z-x)^3=3(x-y)(y-x)(z-x)$$

Question

Show that 
$$(x-y)^3+(y-z)^3+(z-x)^3=3(x-y)(y-x)(z-x)$$

Uniteasy Admin · Accepted Answer

Let
$$a = x-y$$c
$$b = y-z$$c
$$c = z-x$$c
Addition of the three equations results in 
$$a+b+c = 0$$c
For sum of three cubes, there is the following identity 
$$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$$
Then,
$$a^3+b^3+c^3-3abc = 0$$c
Substituting back in terms of $$x,y,z$$ yields
$$(x-y)^3+(y-z)^3+(z-x)^3=3(x-y)(y-x)(z-x)$$

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