Đáp án:
$=3(z-x-1).(y-z).(x-y+1)$
Giải thích các bước giải:
$(x-y+1)^3+(y-z)^3+(z-x-1)^3$
$=[(x-y+1)^3+(y-z)^3)]+(z-x-1)^3$
$=(x-y+1+y-z).[(x-y+1)^2-(x-y+1).(y-z)+(y-z)^2]+(z-x-1)^3$
$=(x-z+1).[(x-y+1)^2-(x-y+1).(y-z)+(y-z)^2]+(z-x-1)^3$
$=-(z-x-1).[(x-y+1)^2-(x-y+1).(y-z)+(y-z)^2]+(z-x-1)^3$
$=(z-x-1).[-(x-y+1)^2+(x-y+1).(y-z)-(y-z)^2+(z-x-1)^2]$
$=(z-x-1).[-(x^2+y^2+1-2xy+2x-2y)+xy-xz-y^2+yz+y-z-(y^2-2yz+z^2)+z^2+x^2+1-2xz+2x-2z]$
$=(z-x-1).( -x^2-y^2-1+2xy-2x+2y+xy-xz-y^2+yz+y-z-y^2+2yz-z^2+z^2+x^2+1-2xz+2x-2z]$
$=(z-x-1).(-3y^2+3xy+3y-3xz+3yz-3z)$
$=(z-x-1).[ (-3y^2+3yz)+(3xy-3xz)+(3y-3z)]$
$=(z-x-1).[ -3y.(y-z)+3x.(y-z)+3.(y-z)]$
$=(z-x-1).(y-z).(-3y+3x+3)$
$=3(z-x-1).(y-z).(x-y+1)$
