Question

Factors of $(p-q{)}^3+(q-r{)}^3+(r-p{)}^3$ are:

• A. pq(r+p)+qr(p+q)
• B. 3(p-q)(q-r)(r-p)
• C. 3(pq)(qr)(rp)
• D. 3(p+q)(q+r)(r+p)

Answer 1

The correct option is B 3(p-q)(q-r)(r-p)

Let $a=p-q,b=q-randc=r-p$
We see that
$a+b+c=(p-q)+(q-r)+(r-p)=0$
Now, we know that $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$
if $(a+b+c)=0$ then $a^3+b^3+c^3-3abc=0$
$\therefore a^3+b^3+c^3=3abc$
$\Rightarrow (p-q{)}^3+(q-r{)}^3+(r-p{)}^3$
$=3(p-q)(q-r)(r-p)$