What are the factors of $(p^{2} - 2 p q + q^{2}) - r^{2}$ ?

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Solution

We have to factorise $(p^{2} - 2 p q + q^{2}) - r^{2}$ .

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

Now using $a^{2} - b^{2} = (a + b) (a - b)$
$(p - q)^{2} - r^{2} = {(p - q) - r} {(p - q) + r}$
$= (p - q - r) (p - q + r)$

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