Without expanding the determinants, show that`|(1,x,x^2),(1,y,y^2),(1,z,z^2)|=(y-x)(z-y)(z-x)`

Without expanding determinant show that |(xa,yb,zc),(a^2,b^2,c^2),(1,1,1)|=|(x,y,z),(a,b,c),(bc,ca,ab)|

Without expanding,show that the values of the following determinants are zero: |(1,x,x^2-yz),(1,y,y^2-zx),(1,z,z^2-xy)|

Expand the following determinants: |(0,x,y),(-x,0,z),(-y,-z,0)|

By using properties of determinant prove that |(x+y,y+z,z+x),(z,x,y),(1,1,1)|=0

Prove that |(x+y,y+z,z+x),(z+x,x+y,y+z),(y+z,z+x,x+y)|=2|(x,y,z),(z,x,y),(y,z,x)|

Without expanding,show that the values of the following determinants are zero: |(1,4,y+z),(1,4,z+x),(1,4,x+y)|

Using properties of determinant show that |(1,log_x y,log_x z),(log_y x,1,log_y z),(log_z x,log_z y,1)|=0

Without expanding,show that the values of the following determinants are zero: |(x,y,z),(x+3a,y+3b,z+3c),(a,b,c)|