Simplify

We need to simplify

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

Step 1: Use identity

We use the identity:

If $a + b + c = 0$, then
$ a^3 + b^3 + c^3 = 3abc $

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

Now,

$ a + b + c = (p-q)+(q-r)+(r-p) = 0 $

So the identity applies.

Step 2: Apply the identity

$ (p - q)^3 + (q - r)^3 + (r - p)^3 = 3(p - q)(q - r)(r - p) $

Step 3: Substitute into the expression

$ \frac{3(p - q)(q - r)(r - p)}{(p - q)(q - r)(r - p)} = 3 $