a³(b - c) + b³(c - a) + c³(a - b)
= a³(b - c) + b³(c - a + b - b) + c³(a - b)
= a³(b - c) + b³[ (c - b) + (b - a) ] + c³(a - b)
= a³(b - c) + b³(c - b) + b³(b - a) + c³(a - b)
= a³(b - c) - b³(b - c) - b³(a - b) + c³(a - b)
= [ a³(b - c) - b³(b - c) ] + [ c³(a - b) - b³(a - b) ]
= (b - c)(a³ - b³) + (a - b)(c³ - b³)
= (b - c)(a - b)(a² + ab + b²) + (a - b)(c - b)(c² + cb + b²)
= (b - c)(a - b)(a² + ab + b²) - (a - b)(b - c)(c² + cb + b²)
= (b - c)(a - b)[ (a² + ab + b²) - (c² + cb + b²) ]
= (b - c)(a - b)(a² + ab + b² - c² - cb - b²)
= (b - c)(a - b)(a² + ab - c² - cb)
= (b - c)(a - b)[ (a² - c²) + (ab + cb) ]
= (b - c)(a - b)[ (a - c)(a + c) + b(a + c) ]
= (b - c)(a - b)[ (a - c)(a + c + b) ]
= (b - c)(a - b)(a - c)(a + c + b)
Vậy a³(b - c) + b³(c - a) + c³(a - b) = (b - c)(a - b)(a - c)(a + c + b)
