x²(y - z) + y²(z - x) + z²(x - y)

= z²(x - y) + x² y - x² z + y² z - y² x

= z²(x - y) + (x² y  - y² x) + (- x² z + y² z)

= (x - y)(z² + xy - zx - zy)

= (x - y)[(z² - zx) + (xy - zy)]

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

\(x^2\left(y-z\right)+y^2\left(z-x\right)+z^2\left(x-y\right)\)

\(=x^2\left(y-z\right)+y^2\left(z-y+y-x\right)\)

\(=x^2\left(y-z\right)-y^2\left(y-z\right)-y^2\left(x-y\right)+z^2\left(x-y\right)\)

\(=\left(x-y\right)\left(x+y\right)\left(y-z\right)+\left(z-y\right)\left(y+z\right)\left(x-y\right)\)

\(=\left(x-y\right)\left(y-x\right)\left(x-z\right)\)

x^2y+xy^2+x^2z+xz^2+y^2z+yz^2+2xyz 
=x^2y+xy^2+xyz+x^2z+xz^2+xyz+y^2z+yz^2 
=xy(x+y+z)+zx(x+y+z)+yz(y+z) 
=x(y+z)(x+y+z)+yz(y+z) 
=(y+z)(x^2+xy+zx+yz) 
=(x+y)(y+z)(z+x)

Ta có: \(x\left(y^2-z^2\right)+y\left(z^2-x^2\right)+z\left(x^2-y^2\right)\)

\(=x\left(y-z\right)\left(y+z\right)+yz^2-x^2y+zx^2-y^2z\)

\(=x\left(y-z\right)\left(y+z\right)-\left(y^2z-yz^2\right)-\left(x^2y-zx^2\right)\)

\(=x\left(y-z\right)\left(y+z\right)-yz\left(y-z\right)-x^2\left(y-z\right)\)

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

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

\(=\left(y-z\right)\left[z\left(x-y\right)-x\left(x-y\right)\right]\)

\(=\left(x-y\right)\left(y-z\right)\left(z-x\right)\)

\(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-x^2z^2\left(z-x\right)\)

\(=x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-x^2z^2\left[\left(z-y\right)+\left(y-x\right)\right]\)

\(=x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-x^2z^2\left(z-y\right)-x^2z^2\left(y-x\right)\)

\(=\left(y-x\right)\left(x^2y^2-x^2z^2\right)+\left(z-y\right)\left(y^2z^2-x^2z^2\right)\)

\(=x^2\left(y-x\right)\left(y-z\right)\left(y+z\right)+z^2\left(z-y\right)\left(y-x\right)\left(y+x\right)\)

\(=\left(y-x\right)\left(z-y\right)\left(-x^2y-x^2z+z^2y+z^2x\right)\)

\(=\left(y-x\right)\left(z-y\right)\left[xz\left(z-x\right)+y\left(z-x\right)\left(z+x\right)\right]\)

\(=\left(y-x\right)\left(z-y\right)\left(z-x\right)\left(xy+yz+xz\right)\)

x(y+z)^2 - y(z-x)^2 +z(x+y)^2 - x^3 + y^3 - z^3 - 4xyz

=xy^2+2xyz+xz^2-yz^2+2xyz-x^2y+x^2z+2xyz+zy^2-x^3+y^3-z^3-4xyz

=xy^2+xz^2-yz^2-x^2y+x^2z+y^2z-x^3+y^3-z^3+2xyz

=(xy^2+2xyz+xz^2)-x^3-(yz^2+2xyz+x^2y)+y^3+(x^2z+2xyz+y^2z)-z^3

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

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

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

=(x+y+z)[-x(x-y-z)-y(x-y-z)+z(x-y-z)]

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

\(x\left(y^2-z^2\right)+y\left(z^2-x^2\right)+z\left(x^2-y^2\right)\)

\(=xy^2-xz^2+yz^2-x^2y+zx^2-zy^2\)

\(=xy^2-xz^2+yz^2-x^2y+zx^2-zy^2-xyz+xyz\)

\(=\left(yz^2-xz^2-xyz+x^2z\right)-\left(zy^2-xyz-xy^2+x^2y\right)\)

\(=z\left(yz-xz-xy+x^2\right)-y\left(zy-xz-xy+x^2\right)\)

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

\(=\left(z-y\right)\left[y\left(z-x\right)-x\left(z-x\right)\right]\)

\(=\left(z-y\right)\left(y-x\right)\left(z-x\right)\)

x(y+z)^2 - y(z-x)^2 +z(x+y)^2 - x^3 + y^3 - z^3 - 4xyz

=xy^2+2xyz+xz^2-yz^2+2xyz-x^2y+x^2z+2xyz+zy^2-x^3+y^3-z^3-4xyz

=xy^2+xz^2-yz^2-x^2y+x^2z+y^2z-x^3+y^3-z^3+2xyz

=(xy^2+2xyz+xz^2)-x^3-(yz^2+2xyz+x^2y)+y^3+(x^2z+2xyz+y^2z)-z^3

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

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

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

=(x+y+z)[-x(x-y-z)-y(x-y-z)+z(x-y-z)]

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