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Bài làm:
Ta có: \(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2+2\left(x-y+z\right)\left(y-z\right)+\left(y-z\right)^2\)(hằng đẳng thức đầu)
\(=\left(x-y+z+y-z\right)^2=x^2\)
\(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2+2\left(x-y+z\right)\left(y-z\right)+\left(y-z\right)^2\)
\(=\left[\left(x-y+z\right)+\left(y-z\right)\right]^2=\left(x-y+z+y-z\right)^2=x^2\)
\(a,\left(x+y\right)^2+\left(x-y\right)^2=x^2+2xy+y^2+x^2-2xy+y^2=2\left(x^2+y^2\right)\)\(b,2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2+\left(x-y\right)^2=2x^2-2y^2+x^2+2xy+y^2+x^2-2xy+y^2=3x^2\)\(c,\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)=\left[\left(x-y+z\right)-\left(z-y\right)\right]^2=\left(x-2y\right)^2\)
a) \(\left(x+y\right)^2+\left(x-y\right)^2\)
=\(\left(x^2+2xy+y^2\right)+\left(x^2-2xy+y^2\right)\)
=\(x^2+2xy+y^2+x^2-2xy+y^2\)
\(2x^2+2y^2=2\left(x^2+y^2\right)\)
b) \(2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2+\left(x-y\right)^2\)
\(=\left(x-y\right)^2+2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2\)
=\(\left[\left(x-y\right)+\left(x+y\right)\right]^2\)
= \(\left(x-y+x+y\right)^2\)
\(=2x^2\)
c) \(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=\left(x-y+z\right)^2-2\left(x-y+z\right)\left(z-y\right)+\left(z-y\right)^2\)
\(=\left[\left(x-y+z\right)-\left(z-y\right)\right]^2\)
= \(\left(x-y+z-z+y\right)^2=x^2\)
\(\left(x+y-z\right)^2+2\left(z-x-y\right)\left(x+y\right)+\left(x+y\right)^2\)
\(=\left(x+y-z\right)^2-2\left(x+y-z\right)\left(x+y\right)+\left(x+y\right)^2\)
\(\left[\left(x+y-z\right)-\left(x+y\right)\right]^2=z^2\)
quy đồng mẫu thức ta được
\(\frac{yz\left(z-y\right)+xz\left(x-z\right)+xy\left(y-x\right)}{xyz\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)\(=\frac{yz\left(z-y\right)+xz\left(x-z\right)-xy\left[\left(z-y\right)+\left(x-z\right)\right]}{xyz\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)
\(=\frac{y\left(z-y\right)\left(z-x\right)+x\left(x-z\right)\left(z-y\right)}{xyz\left(x-y\right)\left(y-z\right)\left(z-x\right)}=\frac{\left(z-y\right)\left(z-x\right)\left(y-x\right)}{xyz\left(z-y\right)\left(z-x\right)\left(y-x\right)}=\frac{1}{xyz}\)
Dat (x-y)2+(y-z)2+(x-z)2=A
=(x+y)3+z3-3x2y-3xy2-3xyz / A
=(x+y+z).(x2+2xy+y2-xy-yz+z2)-3xy(x+y+z) / A
=(x+y+z).(x2+y2+z2-xy-yz-xz) /A
=2(x+y+z).(x2+y2+z2-xy-yz-xz) /2A
=(x+y+z)[ (x2-2xy+y2)+(y2-2yz+z2)+(x2-2xz+z2) / 2A
=(x+y+z).[ (x-y}2+(y-z)2+(x-z)2 ] /2A
=(x+y+z). A /2A
=x+y+z /2
a ) ( x + y )2 +( x - y )2 = x2 + 2xy +y2 + x2 - 2xy + y2
= 2x2 + 2y2
b ) 2 . ( x - y ) . ( x + y ) + ( x + y )2 + ( x - y )2
= 2 . ( x2 - y2 ) + x2 + 2xy + y2 + x2 - 2xy + y2
= 2x2 - 2y2 + x2 +2xy + y2 + x2 - 2xy + y2
= 4x2
c ) ( x - y + z )2 - ( z - y )2 + 2.( x - y + z ) ( y - z )
= x2 + y2 + z2 - 2xy + 2 xz - 2yz - z2 + 2zy - y2 + 2xy - y2 + 2yz -2xz + 2y2 - 2z2
= x2
ngu the
\(\left(x-y+z\right)^2+\left(z-y\right)^2+2\left(x-y+z\right)\left(y-z\right)\)
\(=x^2+y^2+z^2-2xy-2yz+2xz+z^2-2yz+y^2+\left(2y-2z\right)\left(x-y+z\right)\)
\(=x^2+y^2+z^2-2xy-2yz+2xz+z^2-2yz+y^2+2xy-2y^2+2yz-2xz+2yz-2z^2\)
\(=x^2\)
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