Question

Find the factors of $a(b-c{)}^3+b(c-a{)}^3+c(a-b{)}^3$.

Answer 1

Given equation can be written as

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

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