Question

Q.10 Factorize :- (P-Q)3 + (Q-R)3 +
(R-P)3?​

Answer 1

We know the corollary: if a+b+c=0 then a

3

+b

3

+c

3

=3abc

Using the above corollary taking a=p(q−r), b=q(r−p) and c=r(p−q), we have a+b+c=p(q−r)+q(r−p)+r(p−q)=pq−pr+qr−pq+pr−qr=0 then the equation p

3

(q−r)

3

+q

3

(r−p)

3

+r

3

(p−q)

3

can be factorised as follows:

p

3

(q−r)

3

+q

3

(r−p)

3

+r

3

(p−q)

3

=3[p(q−r)×q(r−p)×r(p−q)]=3pqr(q−r)(r−p)(p−q)

Hence, p

3

(q−r)

3

+q

3

(r−p)

3

+r

3

(p−q)

3

=3pqr(q−r)(r−p)(p−q)