Question

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a^3(b-c)+b^3(c-a)+c^3(a-b)

Answer 1

Ответ: (a - c)(b - c)(a - b)(a + b + c).

Пошаговое объяснение:

a³(b - c) + b³(c - a) + c³(a - b) = а³b - a³c + b³c - ab³ + ac³ - bc³ =

= (a³b - bc³) - (a³c - ac³) + (b³c - b³a) = b(a³ - c³) - ac(a² - c²) - b³(a - c) =

= b(a - c)(a² + ac + c²) - ac(a - c)(a + c) - b³(a - c) = (a - c)(b(a² + ac + c²) -

- ac(a + c) - b³) = (a - с)(a²b + abc + bc² - a²c - ac² - b³) = (a - c) ×

× ((a²b - a²c) + (abc - ac²) + (bc² - b³)) = (a - c)(a²(b - c) + ac(b - c) - b ×

× (b² - c²) = (a - c)(b - c)(a² + ac - b(b + c)) = (a - c)(b - c)(a² + ac - b² - bc) =

= (a - c)(b - c)(a² - b² + ac - bc) = (a - c)(b - c)((a - b)(a + b) + c(a - b)) =

= (a - c)(b - c)(a - b)(a + b + c).