Question

prove that $a^3{\left(b-c\right)}^3+b^3{\left(c-a\right)}^3+c^3{\left(a-b\right)}^3=$ $a^3(c-b)+b^3(a-c)+c^3(b-a)$

Answer 1

We have,

$a{(b-c)}^3+b{(c-a)}^3+c{(a-b)}^3$

$\Rightarrow a(b^3-c^3-3bc(b-c))+b(c^3-a^3-3ac(c-a))+c(a^3-b^3-3ab(a-b))$

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

$\Rightarrow a{b}^3-a{c}^3-3a{b}^{2}c+3ab{c}^2+b{c}^3-b{a}^3-3ab{c}^2+3a^{2}bc+c{a}^3-c{b}^3-3a^{2}bc+3a{b}^{2}c$

$\Rightarrow a{b}^3-a{c}^3+b{c}^3-b{a}^3+c{a}^3-c{b}^3$

$\Rightarrow a^3(c-b)+b^3(a-c)+c^3(b-a)$

Hence proved.