Factorise : a3 (b-c)3 + b3 (c-a)3 +c3 (a-b)3 - Maths - Polynomials

Question

Factorise :
a3 (b-c)3 + b3
(c-a)3 +c3 (a-b)3

Aakash Sharma · Accepted Answer

$a^{3} (b - c)^{3} + b^{3} (c - a)^{3} + c^{3} (a - b)^{3} = {[a (b - c)]}^{3} + [b (c - a)]^{3} + [c (a - b)]^{3}$

We know that if a + b + c =0 then $a^{3} + b^{3} + c^{3} = 3 a b c$

$\Rightarrow a + b + c = [a (b - c)] + b (c - a) + c (a - b) = a b - a c + b c - a b + a c - b c = 0 ∴ {[a (b - c)]}^{3} + {[b (c - a)]}^{3} + {[c (a - b)]}^{3} = 3 [a (b - c)] [b (c - a)] [c (a - b)] = 3 a b c (b - c) (c - a) (a - b)$