\(y+z=-x\)
\(\left(y+z\right)^5=-x^5\)
\(y^5+5y^4z+10y^3z^2+10y^2z^3+5yz^4+z^5+x^5=0\)
\(x^5+y^5+z^5+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(x^5+y^5+z^5+5yz\left(\left(y+z\right)\left(y^2-yz+z^2\right)+2yz\left(y+z\right)\right)=0\)
\(x^5+y^5+z^5+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)-5xyz\left(\left(y^2+2yz+z^2\right)+y^2+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)
Ta có: \(y+z=-x\)
\(\left(y+z\right)^5=-x^5\)
\(y^5+5y^4z+10y^3z^2+10y^2z^3+5yz^4+z^5+x^5=0\)
\(x^5+y^5+z^5+5yz\left(y^3+2y^2z+2yz^2+z^3\right)=0\)
\(x^5+y^5+z^5+5yz\left(\left(y+z\right)\left(y^2-yz+z^2\right)+2yz\left(y+z\right)\right)=0\)
\(x^5+y^5+z^5+5yz\left(y+z\right)\left(y^2+yz+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)-5xyz\left(\left(y^2+2yz+z^2\right)+y^2+z^2\right)=0\)
\(2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\)
Ta có: \(x+y+z=0\Rightarrow x+y=-z\Rightarrow\left(x+y\right)^3=\left(-z\right)^3\Rightarrow x^3+y^3+3xy\left(x+y\right)=-z^3\Rightarrow x^3+y^3+z^3=-3xy\left(x+y\right)=-3xy.\left(-z\right)=3xyz\Rightarrow\left(x^2+y^2+z^2\right)\left(x^3+y^3+z^3\right)=3xyz\left(x^2+y^2+z^2\right)\)\(\Leftrightarrow x^5+y^5+z^5+x^3\left(y^2+z^2\right)+y^3\left(z^2+x^2\right)+z^3\left(x^2+y^2\right)=3xyz\left(x^2+y^2+z^2\right)\Leftrightarrow x^5+y^5+z^5+x^3\left[\left(y+z\right)^2-2yz\right]+y^3\left[\left(z+x\right)^2-2zx\right]+z^3\left[\left(x+y\right)^2-2xy\right]=3xyz\left(x^2+y^2+z^2\right)\)\(\Leftrightarrow x^5+y^5+z^5+x^3\left[x^2-2yz\right]+y^3\left[y^2-2zx\right]+z^3\left[z^2-2xy\right]=3xyz\left(x^2+y^2+z^2\right)\Leftrightarrow2\left(x^5+y^5+z^5\right)=5xyz\left(x^2+y^2+z^2\right)\left(đpcm\right)\)
