Question

Докажите правильность.

1) (a+b)³=a(a-3b)²+b(b-3a)²
2)(a³+b³)²-(a²+b²)³+3a²b²(a+b)²=(2ab)³
Зарание спасибо.

Answer 1

1)
$(a + b)^{3} = {a}^{3} + 3 {a}^{2} b + 3a {b}^{2} + {b}^{3}$
$a {(a - 3b)}^{2} +b {(b - 3a)}^{2} = a( {a}^{2} - 6ab + 9 {b}^{2} ) +b( {b}^{2} - 6ab + 9 {a}^{2} ) = {a}^{3} - 6 {a}^{2} b + 9a {b}^{2} + {b}^{3} - 6a {b}^{2} + 9 {a}^{2} b = {a}^{3} + 3 {a}^{2} b + 3a {b}^{2} + {b}^{3}$
ч.т.д.

2)
$( {a}^{3} + {b}^{3} )^{2} - ( {a}^{2} + {b}^{2} )^{3} + 3 {a}^{2} {b}^{2} (a + b)^{2} = {a}^{6} + 2 {a}^{3} {b}^{3} + {b}^{6} - {a}^{6} - 3 {a}^{4} {b}^{2} - 3 {a}^{2} {b}^{4} - {b}^{6} + 3 {a}^{4} {b}^{2} + 6 {a}^{3} {b}^{3} + 3 {a}^{2} {b}^{4} = 8 {a}^{3} {b}^{3} = (2ab)^{3}$
ч.т.д.