Question

Cho x^2 y - y^2 x + x^2 z - z^2 x + y^2 z + z^2 y = 2xyz
Cmr trong 3 số x,y,z ít nhất có 2 số = nhau hoặc đối nhau ?

Answer 1

\(x^2y-y^2x+x^2z-z^2x+y^2z+z^2y=2xyz\)\(\Leftrightarrow\left(x^2y-xy^2\right)+\left(x^2z-xyz\right)+\left(z^2y-z^2x\right)+\left(y^2z-xyz\right)=0\)\(\Leftrightarrow xy\left(x-y\right)+xz\left(x-y\right)-z^2\left(x-y\right)-yz\left(x-y\right)=0\)\(\Leftrightarrow\left(xy+xz-z^2-yz\right)\left(x-y\right)=0\)

\(\Leftrightarrow\left(x-y\right)\left[x\left(y+z\right)-z\left(y+z\right)\right]=0\)

\(\Leftrightarrow\left(x-y\right)\left(x-z\right)\left(y+z\right)=0\Rightarrow\left[{}\begin{matrix}x-y=0\\x-z=0\\y+z=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=y\\x=z\\y=-z\end{matrix}\right.\Rightarrowđpcm\)

Answer 2

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

\(x^2y-y^2x+x^2z-z^2x+y^2z+z^2y-2xyz=0\)

\(\left(x^2y-y^2x\right)+\left(x^2z-xyz\right)+\left(z^2y-z^2x\right)=\left(y^2z-xyz\right)+\left(y^2z-xyz\right)=0\)

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

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

đpcm