Phân tích đa thức thành nhân tử: (x + y + z)^3 – x^3 – y^3 – z^3

* Hướng dẫn giải

$$\begin{gathered} {\left(x+y+z\right)}^3-z^3-y^3-z^3 \\ = {\left[(x + y) + z\right]}^3 – x^3 – y^3 – z^3 \\ = {(x + y)}^3 + 3{(x + y)}^{2}z + 3(x + y)z^2 + z^3 – x^3 – y^3 – z^3 \\ = x^3 + y^{3 }+ 3xy(x + y) + 3{(x + y)}^{2}z + 3(x + y)z^2 – x^3 – y^3 \\ ( vì z^3 – z^3 = 0 ; 3x^{2}y + 3x{y}^2 = 3xy (x + y) ) \\ = 3xy.(x+ y) + 3{( x+ y)}^2.z + 3(x+ y).z^2 \\ = 3(x + y)[xy + (x + y)z + z^2] \\ = 3(x + y)[xy + xz + yz + z^2] \\ = 3(x + y)\left[x\right(y + z) + z(y + z\left)\right] \\ = 3(x + y)(y + z)(x + z) \end{gathered}$$