вправа 731 гдз 7 клас алгебра Мерзляк Полонський

Відповідь:

1) (a + b + c)³ - a³ - b³ - c³ =
= ((a + b + c)³ - a³) - (b³ + c³) =
= (a + b + c - a)((a + b + c)² +
+ a(a + b + c) + a²) -
- (b + c)(b² - bc + c²) =
= (b + c)(a² + b² + c² + 2ab + 2ac +
+ 2bc + a² + ab + ac + a² - b² + bc - c²) =
= (b + c)(3a² + 3ab + 3ac + 3bc) =
= 3(b + c)(a(a + b) + c(a + b)) =
= 3(a + b)(a + c)(b + c);
2) (a - b)³ + (b - c)³ - (a - c)³ =
= (a³ - 3a²b + 3ab² - b³) + (b³ - 3b²c +
+ 3bc² - c³) - (a³ - 3a²c + 3ac² - c³) =
= a³ - 3a²b + 3ab² - b³ + b³ - 3b²c +
+ 3bc² - c³ - a³ + 3a²c - 3ac² + c³ =
= -3a²b + 3ab² - 3b²c +
+ 3bc² + 3a²c - 3ac² =
= -3(a²b - ab² + b²c - bc² + ac² - a²c) =
= -3(ab(a - b) + c²(a - b) - c(a² - b²)) =
= -3(ab(a - b) +
+ c²(a - b) - c(a - b)(a + b)) =
= -3(a - b)(ab + c² - c(a + b)) =
= -3(a - b)(ab + c² - ac - bc) =
= -3(a - b)(ab - bc + c² - ac) =
= -3(a - b)(b(a - c) - c(a - c)) =
= -3(a - b)(b - c)(a - c).