Question

Simplify: $\frac{\left(a^2-b^2\right)+{\left(b^2-c^2\right)}^3+{\left(c^2-a^2\right)}^3}{{\left(a-b\right)}^3+{\left(b-c\right)}^3+{\left(c-a\right)}^3}$

Answer 1

If $a+b+c=0,$ then $a^3+b^3+c^3=3abc$

$\left(\frac{{(a^2-b^2)}^3+{(b^2-c^2)}^3+{(c^2-a^2)}^3}{{(a-b)}^3+{(b-c)}^3+{(c-a)}^3}\right)$

$\left(\frac{{(a^2-b^2)}^3+{(b^2-c^2)}^3+{(c^2-a^2)}^3}{{(a-b)}^3+{(b-c)}^3+{(c-a)}^3}\right)$

$\left(\frac{(a-b)(a+b)(b-c)(b+c)(c-a)(c+a)}{(a-b)(b-c)(c-a)}\right)$

$(a+b)(b+c)(c+a)$