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

Question

Simplify

\(\cfrac{{(a^2-b^2)}^3+{(b^2-c^2)}^3+{(c^2-a^2)}^3}{{(a-b)}^3+{(b-c)}^3{(c-a)}^3}\)

A) 1 

B) 3(a + b) (b + c) (c + a) 

C) (a + b) (b + c) (c + a) 

D) 2(a + b) (b + c) (c + a)

Answer 1

Correct option is (C) (a + b) (b + c) (c + a)

If a+b+c = 0, then \(a^3+b^3+c^3=3abc\)

\(\therefore\) \((a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3\) \(=3(a^2-b^2)(b^2-c^2)(c^2-a^2)\)

\((\because(a^2-b^2)+(b^2-c^2)+(c^2-a^2)=0)\)

and \((a-b)^3+(b-c)^3+(c-a)^3\) \(=3(a-b)(b-c)(c-a)\)

\((\because(a-b)+(b-c)+(c-a)=0)\)

\(\therefore\) \(\frac{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}{(a-b)^3+(b-c)^3+(c-a)^3}\) \(=\frac{3(a^2-b^2)(b^2-c^2)(c^2-a^2)}{3(a-b)(b-c)(c-a)}\)

= (a+b) (b+c) (c+a)

Answer 2

Correct option is C) (a + b) (b + c) (c + a)