Find the factors of $a (b - c)^{2} + b (c - a)^{2} + c (a - b)^{2} + 8 a b c$ .

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Solution

Here,
$p (x) = a (b - c)^{2} + b (c - a)^{2} + c (a - b)^{2} + 8 a b c$
$= a (b^{2} + c^{2} - 2 b c) + b (c^{2} + a^{2} - 2 a c) + c (a^{2} + b^{2} - 2 a b) + 8 a b c$
$= a b^{2} + a c^{2} - 2 a b c + b c^{2} + b a^{2} - 2 a b c + c a^{2} + c b^{2} - 2 a b c + 8 a b c$
$= a b^{2} + a c^{2} + b c^{2} + b a^{2} + c a^{2} + c b^{2} + 2 a b c$
$= (b + c) (c + a) (a + b)$
Therefore, $(b + c) (c + a) (a + b)$ is a factor of $a (b - c)^{2} + b (c - a)^{2} + c (a - b)^{2} + 8 a b c .$

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