Question

Express the following in factors from :
$a^3({b-c)}^3+b^3{(c-a)}^3+c^3{(a-b)}^3$

Answer 1

Given

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

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

Let,

$a(b-c)=x$

$b(c-a)=y$

$c(a-b)=z$

Here,

$x+y+z=a(b-c)+b(c-a)+c(a-b)$

$=ab=ac+bc-ab+ac-bc$

=0

Thus,

$a^3(b-c{)}^3+b^3(c-a{)}^3+c^3(a-b{)}^3=x^3+y^3+z^3$

$=3xyz$

$=3\left(a\right(b-c\left)b\right(c-a\left)c\right(a-b\left)\right)$

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