phân tích đa thức thành nhân tử (x-y+1)^3+(y-z)^3+(z-x-1)^3

Đáp án:

$=3(z-x-1).(y-z).(x-y+1)$

Giải thích các bước giải:

 $(x-y+1)^3+(y-z)^3+(z-x-1)^3$

$=[(x-y+1)^3+(y-z)^3)]+(z-x-1)^3$

$=(x-y+1+y-z).[(x-y+1)^2-(x-y+1).(y-z)+(y-z)^2]+(z-x-1)^3$

$=(x-z+1).[(x-y+1)^2-(x-y+1).(y-z)+(y-z)^2]+(z-x-1)^3$

$=-(z-x-1).[(x-y+1)^2-(x-y+1).(y-z)+(y-z)^2]+(z-x-1)^3$

$=(z-x-1).[-(x-y+1)^2+(x-y+1).(y-z)-(y-z)^2+(z-x-1)^2]$

$=(z-x-1).[-(x^2+y^2+1-2xy+2x-2y)+xy-xz-y^2+yz+y-z-(y^2-2yz+z^2)+z^2+x^2+1-2xz+2x-2z]$

$=(z-x-1).( -x^2-y^2-1+2xy-2x+2y+xy-xz-y^2+yz+y-z-y^2+2yz-z^2+z^2+x^2+1-2xz+2x-2z]$

$=(z-x-1).(-3y^2+3xy+3y-3xz+3yz-3z)$

$=(z-x-1).[ (-3y^2+3yz)+(3xy-3xz)+(3y-3z)]$

$=(z-x-1).[ -3y.(y-z)+3x.(y-z)+3.(y-z)]$

$=(z-x-1).(y-z).(-3y+3x+3)$

$=3(z-x-1).(y-z).(x-y+1)$