Question

Докажите тождество: (2a^3 - 3a^2b + 3ab^2 - 2b^3)^2 - 9a^2b^2*(a - b)^2=4*(a - b)^4*(a^2 + ab + b^2). (a -3b)^3 - (2a - 3b)*((3ab + (a - 3b)^2) = -a^3

Answer 1

1
[2(a³-b³)-3ab(a-b)]²-9a²b²(a-b)²=[2(a-b)(a²+ab+b²)-3ab(a-b)]²-9a²b²(a-b)²=
=[(a-b)(2a²+2ab+2b²-3ab)]²-9a²b²(a-b)²=(a-b)²(2a²-ab+2b²)²-9a²b²(a-b)²=
=(a-b)²(2a²-ab+2b²-3ab)(2a²-ab+b²+3ab)=(a-b)²(2a²-4ab+2b²)(2a²+2ab+b²)=
=4(a-b)²(a²-2ab+b²)(a²+ab+b²)=4(a-b)²(a-b)²(a²+ab+b²)=4(a-b)^4*(a²+ab+b²)

4(a-b)^4*(a²+ab+b²)=4(a-b)^4*(a²+ab+b²)

2
(a -3b)³ - (2a - 3b)*(3ab + (a - 3b)² =(a-3b)²-(2a-3b)(3ab+a²-6ab+9b²)=
=(a-3b)²-(2a-3b)(a²+3ab+9b²)=a³-9a²b+27ab²-27b²-2a³+6a²b-18ab²+3a²b-9ab²=27b³=-a3

-a³=-a³