Question

已知a+b+c+d=0,a^3+b^3+c^3+d^3=3 (1)求证;(a+b)^3+(c+d)^3=0 (2)求证:

Answer 1

(1)

a+b+c+d=0

a+b = -c-d

(a+b)² = (c+d)²

(a+b)³+(c+d)³

= (a+b)(a+b)²+(c+d)³

= (a+b)(c+d)²+(c+d)³

= (a+b+c+d)(c+d)²

= 0*(c+d)²

= 0

(2)

(a+b)³+(c+d)³=0

a³+3a²b+3ab²+b³ + c³+3c²d+3cd²+d³=0

a³+b³+c³+d³+3a²b+3ab²+3c²d+3cd²=0

3+3a²b+3ab²+3c²d+3cd²=0

1+a²b+ab²+c²d+cd²=0

1+ab(a+b)+cd(c+d)=0

1-ab(c+d)-cd(a+b)=0

∴ab(c+d)+cd(a+b)=1

不好意思,刚才忘了答第二问