a/ \(x^2-5x+5y-y^2=\left(x^2-y^2\right)-\left(5x-5y\right)=\left(x-y\right)\left(x+y\right)-5\left(x-y\right)=\left(x-y\right)\left(x+y-5\right)\)
b/ \(3x^2-6xy+3y^2-12z^2=3\left(x^2-2xy+y^2-4z^2\right)=3\left[\left(x^2-2xy+y^2\right)-\left(2x\right)^2\right]=3\left[\left(x-y\right)^2-\left(2x\right)^2\right]=3\left(x-y-2x\right)\left(x-y+2x\right)=3\left(-x-y\right)\left(3x-y\right)\)
c/ \(x^2-2xy+y^2-xz+yz=\left(x^2-2xy+y^2\right)-\left(xz-yz\right)=\left(x-y\right)^2-z\left(x-y\right)=\left(x-y\right)\left(x-y-z\right)\)
d/ \(x^2-x+2y-4y^2=\left(x^2-4y^2\right)-\left(x+2y\right)=\left(x+2y\right)\left(x-2y\right)-\left(x+2y\right)=\left(x+2y\right)\left(x-2y-1\right)\)
e/ \(x^6-y^6=\left(x^3\right)^2-\left(y^3\right)^2=\left(x^3-y^3\right)\left(x^3+y^3\right)\)
a) x2 - 5x + 5y - y2
= ( x2 - y2 ) - ( 5x - 5y )
= ( x - y )( x + y ) - 5( x - y )
= ( x - y )( x + y - 5 )
b) 3x2 - 6xy + 3y2 - 12z2
= 3( x2 - 2xy + y2 - 4z2 )
= 3[( x2 - 2xy + y2 ) - 4z2 ]
= 3[( x - y )2 - 4z2 ]
= 3( x - y - 2z )( x - y + 2z )
c) x2 - 2xy + y2 - xz - yz
= ( x2 - 2xy + y2 ) - ( xz - yz )
= ( x - y )2 - z( x - y )
= ( x - y )( x - y - z )
d) x2 - x + 2y - 4y2
= ( x2 - 4y2 ) - ( x - 2y )
= ( x - 2y )( x + 2y ) - ( x - 2y )
= ( x - 2y )(x + 2y - 1 )
e) x6 - y6
= ( x3 )2 - ( y3 )2
= ( x3 - y3 )( x3 + y3 )
= ( x - y )( x2 + xy + y2 )( x + y )( x2 - xy + y2 )
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