Question

Phân tích đa thức thành nhân tử:

(x-y)³ +(y-z)³ + (z-x)³

Answer 1

\(\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3\\ =\left(x-y+y-z\right)\left[\left(x-y\right)^2+\left(x-y\right)\left(y-z\right)+\left(y-z\right)^2\right]+\left(z-x\right)^3\\ =-\left(z-x\right)\left(x^2-2xy+y^2+xy+yz-y^2-xz+y^2-2yz+z^2\right)+\left(z-x\right)^3\\ =-\left(z-x\right)\left(x^2+y^2+z^2-xy-yz-zx\right)+\left(z-x\right)^3\\ =-\left(z-x\right)\left(x^2+y^2+z^2-xy-yz-zx+z^2-2zx+x^2\right)\\ =-\left(z-x\right)\left(2x^2+y^2+2z^2-xy-yz-3zx\right)\)

Answer 2

(x-y)³+(y-z)³+(z-x)³

=(x-y)³+(y-x+x-z)³+(z-x)³

=(x-y)³+(y-x)³+(x-z)³+3(y-x)(x-z)(y-x+x-z)+(x-z)³

=-(y-x)³+(y-x)³+(x-z)³+3(y-x)(x-x)(x-z)(y-z)-(z-x)³

=3(y-x)(x-z)(y-z)