Factorise the following by taking out the common factors:
(k) $p (p^{2} + q^{2} - r^{2}) + q (r^{2} - q^{2} - p^{2}) - r (p^{2} + q^{2} - r^{2})$

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Solution

Given: $p (p^{2} + q^{2} - r^{2}) + q (r^{2} - q^{2} - p^{2}) - r (p^{2} + q^{2} - r^{2})$

$= p (p^{2} + q^{2} - r^{2}) - q (- r^{2} + q^{2} + p^{2}) - r (p^{2} + q^{2} - r^{2})$

$= p (p^{2} + q^{2} - r^{2}) - q (p^{2} + q^{2} - r^{2}) - r (p^{2} + q^{2} - r^{2})$

The common factor is $(p^{2} + q^{2} - r^{2})$

Dividing by $(p^{2} + q^{2} - r^{2})$ in above expression ,we get,

$\frac{p (p^{2} + q^{2} - r^{2})}{(p^{2} + q^{2} - r^{2})} - \frac{q (p^{2} + q^{2} - r^{2})}{(p^{2} + q^{2} - r^{2})} - \frac{r (p^{2} + q^{2} - r^{2})}{(p^{2} + q^{2} - r^{2})}$

$= p - q - r$

$∴ p (p^{2} + q^{2} - r^{2}) + q (r^{2} - q^{2} - p^{2}) - r (p^{2} + q^{2} - r^{2}) = (p^{2} + q^{2} - r^{2}) (p - q - r)$

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