$1. Cho x+2y+z=0$
$CMR: x^3+8y^3+z^3= ^xyz$
$2. Cho a+b+c+d=0$
$CMR: a^3+b^3+c^3+d^3= 3(c+d)(ab-cd)$
$Giúp mình gấp ạ!!!$
$1. Cho x+2y+z=0$
$CMR: x^3+8y^3+z^3= ^xyz$
$2. Cho a+b+c+d=0$
$CMR: a^3+b^3+c^3+d^3= 3(c+d)(ab-cd)$
$Giúp mình gấp ạ!!!$
`x+2y+z=0`
`→(x+2y+z)^3=0`
`↔[(x+z)+2y]^3=0`
`↔(x+z)^3+6y(x+z)^2+12y^2(x+z)+8y^3=0`
`↔x^3+8y^3+z^3+3x^2z+3xz^2+6y(x+z)(x+2y+z)=0`
`↔x^3+8y^3+z^3+3xz(x+z)=0` (vì `x+2y+z=0`)
`↔x^3+8y^3+z^3=-3xz(x+z)`
`↔x^3+8y^3+z^3=-3xz(-2y)=6xyz`
`a^3 + b^3 + c^3 + d^3`
Theo đề cho, ta có :
`a + b + c + d = 0`
`a + b = – ( c + d )`
Chuyển vế xuống ta sẽ có :
`( a + b )^3 = -( c + d )^3`
Phân tích : `( a + b )^3 = a^3 + b^3 + 3ab( a + b )`
`-( c + d )^3 = -c^3 – d^3 – 3cd(c+d)`
`Có : a + b = – ( c + d )`
`⇒ a^3 + b^3 + c^3 + d^3 = 3ab(c+d) – 3cd(c+d)`
`3ab(c+d) – 3cd(c+d)`
`= ( 3ab – 3cd )(c+d)`
`= 3(ab – cd)(c+d)` hay `3(c+d)(ab-cd) ( đpcm )`