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Question

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


Solution

Consider, c³(a−b)³+a³(b−c)³+b³(c-a)³
= (ac − bc)³ + (ab − ac)³+ (bc - ab)³
Consider, x = (ac − bc), y = (ab − ac) and z = (bc - ab)
⇒ x + y + z = (ac − bc) + (ab − ac) + (bc - ab) = 0
If x + y + z = 0 then x3 + y3 + z3 = 3xyz
Hence (ac − bc)³ + (ab − ac)³+ (bc - ab)³ = 3(ac − bc)(ab − ac)(bc - ab)
= 3c(a − b) × a(b − c) × b(c - a)
= 3abc(a − b)(b − c)(c - a)

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