Q.165. Simplification of algebraic expression

Question 165.
Simplify:
[(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3] / [(a-b)^3+(b-c)^3+(c-a)^3]

Answer 165.
Let
a - b = x,
b - c = y and
c - a = z
=> x + y + z = 0
=> x^3 + y^3 + z^3 = 3xyz
=> (a - b)^3 + (b - c)^3 + (c - a)^3 = 3(a - b) (b - c) (c - a) ... ( 1 )

Similarly, it can be proved that
(a^2 - b^2)^3 + (b^2 - c^2)^3 + (c^2 - a^2)^3 
                         = 3(a^2 - b^2) (b^2 - c^2) (c^2 - a^2) ... ( 2 )

From ( 1 ) and ( 2 ),
[ (a^2-b^2)^3 + (b^2-c^2)^3 + (c^2-a^2)^3 ] /
                                            [(a-b)^3 + (b-c)^3 + (c-a)^3]
= [3(a^2 - b^2) (b^2 - c^2) (c^2 - a^2)] / [3(a - b) (b - c) (c - a)]
= [3(a+b) (a-b) (b+c) (b-c) (c+a) (c-a)] / 3(a-b) (b-c) (c-a)]
= (a + b) (b + c) (c + a).

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Proof of theory used.
x + y + z = 0
=> x + y = - z
=> (x + y)^3 = - z^3
=> x^3 + y^3 + 3xy(x + y) = -z^3
=> x^3 + y^3 + 3xy(-z) = -z^3
=> x^3 + y^3 + z^3 = 3xyz.

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