Using the properties of determinant, show that : `|[1,x+y,x^2+y^2],[1,y+z,y^2+z^2],[1,z+x,z^2+x^2]| = (x-y)(y-z)(z-x)`

Using the properties of determinants, show that : |[[x, y, z],[x^2, y^2, z^2],[x,y,z]]|= 0 .

Using the properties of determinants, show that : |[[x^2, y^2, z^2],[yz, zx, xy],[x,y,z]]|= (x-y)(y-z)(z-x)(xy+yz+zx) .

Using properties of determinants, prove that: |[x,x^2,1+px^3],[y,y^2,1+py^3],[z,z^2,1+pz^3]| = (1+pxyz)(x-y)(y-z)(z-x)

By using properties of determinants, show that : |[x+y+2z,x,y],[z,y+z+2x,y],[z,z,z+x+2y]| = 2(x+y+z)^3

Using the properties of determinants, prove that : |[[a+x,y,z],[x,a+y,z],[x,y,a+z]]|=a^2(a+x+y+z)

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)| = (x-y)(y-z)(z-x)

Prove that: |[x,x^2,yz],[y,y^2,zx],[z,z^2,xy]|=(x-y)(y-z)(z-x)(xy+yz+zx)

Using properties of determinants, prove that : {:|((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3

Using properties of determinants , prove that |(x^2+1,xy,zx),(xy,y^2+1,yz),(zx,yz,z^2+1)|=1+x^2+y^2+z^2

Show that |(1,x^2,x^3),(1,y^2,y^3),(1,z^2,z^3)| = (x-y),(y-z)(z-x)(xy+yz+zx)