\(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=x^2+y^2+z^2-2xy-2yz+2xz+z^2-2yz+y^2+\left(2y-2z\right)\left(x-y+z\right)\)
\(=x^2+y^2+z^2-2xy-2yz+2xz+z^2-2yz+y^2+2xy-2y^2+2yz-2xz+2yz-2z^2\)
\(=x^2\)
Ta có: (x - y + z)2 +2(x - y + z)( y - z) +( z- y)2 = (x - y + z+ z- y)2 =(x - 2y + 2z)2
\(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2+\left(y-z\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z+y-z\right)^2\)
\(=x^2\)
\(\left(x-y+z\right)^2\) \(+\)\(\left(z-y\right)^2\) \(+\)\(2\)\(\left(x-y+z\right)\)\(\left(y-z\right)\)
\(=\)\(x^2\)\(+\)\(y^2\)\(+\)\(z^2\)\(-\)\(2xy\)\(-\)\(2yz+2xz\)\(+\)\(z^2\)\(-\)\(2yz\)\(+\)\(y^2\)\(+\)\(\left(2y-2z\right)\)\(\left(x-y+z\right)\)
\(=\)\(x^2+y^2+z^2-2xy-2yz+2xz+z^2-2yz+y^2+2xy-\)\(2y^2+2yz-2xz+2yz-2z^2\)
\(=\)\(x^2\)
\(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z+x-y\right)^2\)( dùng hằng đẳng thức số 1)
\(=\left(2x-2y+z\right)^2\)
( x - y + z )2 + ( z - y )2 + 2( x - y + z )( y - z )
= [ ( x - y ) + z ]2 + z2 - 2zy + y2 + ( 2x - 2y + 2z )( y - z )
= ( x - y )2 + 2( x - y)z + z2 + z2 - 2zy + y2 + 2xy - 2xz - 2y2 + 2yz + 2zy - 2z2
= x2 - 2xy + y2 + 2xz - 2yz + z2 + z2 - 2zy + y2 + 2xy - 2xz - 2y2 + 2yz + 2zy - 2z2
= x2
