Sửa đề z^4(z-y) thành z^4(x-y)
Đặt \(A=x^4\left(y-z\right)+y^4\left(z-x\right)+z^4\left(x-y\right)\)
\(=x^4\left(y-x+x-z\right)+y^4\left(z-x\right)+z^4\left(x-y\right)\)
\(=-x^4\left(x-y\right)+x^4\left(x-z\right)-y^4\left(x-z\right)+z^4\left(x-y\right)\)
\(=\left(x-y\right)\left(z^4-x^4\right)+\left(x-z\right)\left(x^4-y^4\right)\)
\(=\left(x-y\right)\left(z^2+x^2\right)\left(z^2-x^2\right)+\left(x-z\right)\left(x^2+y^2\right)\left(x^2-y^2\right)\)
\(=\left(x-y\right)\left(z^2+x^2\right)\left(x+z\right)\left(z-x\right)+\left(x-z\right)\left(x^2+y^2\right)\left(x+y\right)\left(x-y\right)\)
\(=\left(x-y\right)\left(z-x\right)\left[\left(z^2+x^2\right)\left(x+z\right)-\left(x^2+y^2\right)\left(x+y\right)\right]\)
\(=\left(x-y\right)\left(z-x\right)\left(xz^2+z^3+x^3+x^2z-x^3-x^2y-xy^2-y^3\right)\)
\(=\left(x-y\right)\left(z-x\right)\left[x^2\left(z-y\right)+x\left(z^2-y^2\right)+\left(z^3-y^3\right)\right]\)
\(=\left(x-y\right)\left(z-x\right)\left(z-y\right)\left[x^2+x\left(z+y\right)+\left(z^2+yz+y^2\right)\right]\)
\(=\left(x-y\right)\left(x-z\right)\left(y-z\right)\left(x^2+xz+xy+z^2+yz+y^2\right)\)
\(=\frac{1}{2}\left(x-y\right)\left(x-z\right)\left(y-z\right)\left(2x^2+2y^2+2z^2+2xy+2yz+2xz\right)\)
\(=\frac{1}{2}\left(x-y\right)\left(x-z\right)\left(y-z\right)\left[\left(x+y\right)^2+\left(y+z\right)^2+\left(z+x\right)^2\right]\)
Vì \(x>y>z\Rightarrow\hept{\begin{cases}x-y>0\\x-z>0\\y-z>0\end{cases}}\) và \(\left(x+y\right)^2+\left(y+z\right)^2+\left(z+x\right)^2\ge0\)
=>....
