Question

prove that a^3(b-c)^3+b^3 (c-a)^3+c^3 (a-b)^3=3abc (a-b)(b-c)(c-a)

Answer 1

we can write a³(b-c)³+b³(c-a)³+c³(a-b)³ as (a(b-c))³+(b(c-a))³+(c(a-b))³
we know that if a+b+c = 0
a³+b³+c³ = 3abc
⇒ a+b+c =a(b-c)+b(c-a)+c(a-b)
= ab-ac+bc-ab+ac-bc
=0
(a(b-c))³+(b(c-a))³+(c(a-b))³ = 3(a(b-c))(b(c-a))(c(a-b))
= 3abc(a-b)(b-c)(c-a)

LHS = RHS
hence proved !!



Hope it helps.... :)