Question

Factorize:

$p^3(q-r{)}^3+q^3(r-p{)}^3+r^3(p-q{)}^3$

Answer 1

$p^3(q-r{)}^3+q^3(r-p{)}^3+r^3(p-q{)}^3$

$=\left\{p\right(q-r){\}}^3+\{q(r-p){\}}^3+\left\{r\right(p-q){\}}^3$

Let
$p(q-r)=a,q(r-p)=b$ and $r(p-q)=c$

$a+b+c=p(q-r)+q(r-p)+r(p-q)$

$=pq-pr+qr-qp+rp-rq$

$=pq-pr+qr-pq+pr-qr$

$=0$

$\because a^3+b^3+c^3=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$

$\therefore a^3+b^3+c^3=3abc[\because (a+b+c)=0]$

$\Rightarrow \left\{p\right(q-r){\}}^3+\{q(r-p){\}}^3+\left\{r\right(p-q){\}}^3=3p(q-r)\times q(r-p)\times r(p-q)$

$\Rightarrow p^3(q-r{)}^3+q^3(r-p{)}^3+r^3(p-q{)}^3=3pqr(p-q\left)\right(q-r\left)\right(r-p)$