Question

Cho \(x+y+z=0.\)

Chứng minh rằng :

\(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+3xyz=0\)

Answer 1

Ta có : \(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+3xyz\)

\(=\left[xy\left(x+y\right)+xyz\right]+\left[yz\left(y+z\right)+xyz\right]+\left[xz\left(x+z\right)+xyz\right]\)

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

\(=\left(x+y+z\right)\left(xy+yz+xz\right)=0\) (Vì x + y + z = 0 )

Answer 2

\(x+y+z=0\Rightarrow\hept{\begin{cases}x+y=-z\\y+z=-x\\z+x=-y\end{cases}}\)

Từ đó ta có:\(xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)+3xyz\)

\(\Rightarrow xy\left(-z\right)+yz.\left(-x\right)+xz.\left(-y\right)+3xyz\)

\(\Rightarrow-3xyz+3xyz=0\)

\(\Rightarrowđpcm\)