Question

Simplify the following expression:
$\frac{1}{(a-b)(a-c)}+\frac{1}{(b-c)(b-a)}+\frac{1}{(c-a)(c-b)}$

Answer 1

$\frac{1}{(a-b)(a-c)}$ $+\frac{1}{-(c-b)(-(a-b)}$ $+\frac{1}{(-(a-c\left)\right(-(c-b))}$

$=\frac{1}{(a-b)(a-c)}$ $-\frac{1}{(b-c)(a-b)}$ $+\frac{1}{(a-c)(b-c)}$

$=\frac{(b-c+a-c+a-b)}{(a-b)(a-c)(b-c)}$

$=\frac{\left(2\right(a-c\left)\right)}{(a-b)(b-c)(a-c)}$

$=\frac{2}{(a-b)(b-c)}$