a: \(\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2\)
\(=y^2-2yz+z^2+z^2-2xz+x^2+x^2-2xy+y^2\)
\(=2\left(x^2+y^2+z^2-xz-yz-xy\right)\)
Đặt A=x+y+z
\(\left(y+z-2x\right)^2+\left(z+x-2y\right)^2+\left(y+x-2z\right)^2\)
\(=\left(y+z+x-3x\right)^2+\left(z+x+y-3y\right)^2+\left(y+x+z-3z\right)^2\)
\(=\left(A-3x\right)^2+\left(A-3y\right)^2+\left(A-3z\right)^2\)
\(=3A^2-6Ax-6Ay-6Az+9x^2+9y^2+9z^2\)
\(=3A^2-6A\left(x+y+z\right)+9x^2+9y^2+9z^2\)
\(=3A^2-6\left(x+y+z\right)^2+9x^2+9y^2+9z^2\)
\(=3\left(x+y+z\right)^2-6\left(x+y+z\right)^2+9x^2+9y^2+9z^2\)
\(=-3\left(x^2+y^2+z^2+2xy+2yz+2xz\right)+9x^2+9y^2+9z^2\)
\(=6x^2+6y^2+6z^2-6xy-6yz-6xz=6\left(x^2+y^2+z^2-xy-yz-xz\right)\)
Ta có: \(\left(y-z\right)^2+\left(z-x\right)^2+\left(x-y\right)^2=\left(y+z-2x\right)^2+\left(z+x-2y\right)^2+\left(y+x-2z\right)^2\)
=>\(2\left(x^2+y^2+z^2-xz-yz-xy\right)=6\left(x^2+y^2+z^2-xz-yz-xy\right)\)
=>\(4\left(x^2+y^2+z^2-xz-yz-xy\right)=0\)
=>\(2\left(x^2+y^2+z^2-xz-xy-yz\right)=0\)
=>\(2x^2+2y^2+2z^2-2xy-2yz-2xz=0\)
=>\(\left(x^2-2xy+y^2\right)+\left(y^2-2yz+z^2\right)+\left(x^2-2xz+z^2\right)=0\)
=>\(\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2=0\)
=>\(\left\{{}\begin{matrix}x-y=0\\y-z=0\\x-z=0\end{matrix}\right.\Leftrightarrow x=y=z\)
b: \(x\left(x-a\right)\left(x+a\right)\left(x+2a\right)+a^4\)
\(=x\left(x+a\right)\left(x-a\right)\left(x+2a\right)+a^4\)
\(=\left(x^2+ax\right)\left(x^2+ax-2a^2\right)+a^4\)
\(=\left(x^2+ax\right)^2-2a^2\left(x^2+ax\right)+a^4\)
\(=\left(x^2+ax-a^2\right)^2\) là bình phương của một đa thức
