Question

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

Answer 1

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

$={(ab-ac)}^3+{(bc-ba)}^3+{(ca-cb)}^3$

We know that $a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$

$=(ab-ac+bc-ba+ca-cb)({(ab-ac)}^2+{(bc-ba)}^2+{(ca-cb)}^2-(ab-ac)(bc-ba)-(bc-ba)(ca-cb))$

$-(ca-cb)(ab-ac)+3(ab-ac)(bc-ba)(ca-cb)$

$=\left(0\right)({(ab-ac)}^2+{(bc-ba)}^2+{(ca-cb)}^2-(ab-ac)(bc-ba)-(bc-ba)(ca-cb))$

$-(ca-cb)(ab-ac)+3(ab-ac)(bc-ba)(ca-cb)$

$=0+3(ab-ac)(bc-ba)(ca-cb)$

$=3(ab-ac)(bc-ba)(ca-cb)$

$=3a(b-c)b(c-a)c(a-b)$

$=3abc(b-c)(c-a)(a-b)$