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(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(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)

Đặt \(\left\{{}\begin{matrix}a=x+y\\b=y+z\\c=x+z\end{matrix}\right.\Leftrightarrow x+y+z=\dfrac{a+b+c}{2}\)

\(8\left(x+y+z\right)^3-\left(x+y\right)^3-\left(y+z\right)^3-\left(z+x\right)^3\\ =8\left(\dfrac{a+b+c}{2}\right)^3-a^3-b^3-c^3\\ =\left(a+b+c\right)^3-a^3-b^3-c^3\\ =\left(a+b\right)^3+c^3+3\left(a+b\right)c\left(a+b+c\right)-\left(a+b\right)^3+3ab\left(a+b\right)-c^3\\ =3\left(a+b\right)\left(ac+bc+c^2+ab\right)\\ =3\left(a+b\right)\left(b+c\right)\left(a+c\right)\\ =3\left(x+y+y+z\right)\left(y+z+z+x\right)\left(z+x+x+y\right)\\ =3\left(x+2y+z\right)\left(x+y+2z\right)\left(2x+y+z\right)\)

\(z^3\left(x+y^2\right)+y^3\left(z-x^2\right)-x^3\left(y+z^2\right)-xyz\left(xyz-1\right)\)

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

\(=\left(y^2z^3+y^3z\right)+\left(xz^3+xyz\right)-\left(x^2y^3+x^2y^2z^2\right)-x^3\left(y+z^2\right)\)

\(=y^2z\left(y+z^2\right)+xz\left(y+z^2\right)-x^2y^2\left(y+z^2\right)-x^3\left(y+z^2\right)\)

\(=\left(y+z^2\right)\left(y^2z+xz-x^2y^2-x^3\right)\)

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

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

Tick hộ nha bạn 😘

z^3(x+y^2)+y^3(z-x^2)-x^3(y+z^2)-xyz(xyz-1)

\(x^3+y^3+z^3-3xyz\)

\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)

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

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

Bài 3) 

Ta có :

\(x^3+y^3+z^3-3xyz\)

\(\Rightarrow\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)

\(\Rightarrow\left(x+y+z\right)\left[\left(x+y^2\right)-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)

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

P/s tham khảo nha

hok tốt