`|[1,yz,y+z],[1,zx,z+x],[1,xy,x+y]|=|[1,x,x^(2)],[1,y,y^(2)],[1,z,z^(2)]|`

show tha |[1,x,x^2-yz],[1,y,y^2-zx],[1,z,z^2-xy]| =0

solve |[1,yz,yz(y+z)],[1,zx,zx(z+x)],[1,xy,xy(x+y)]|

proof |[x,y,z],[x^(2),y^(2),z^(2)],[yz,zx,xy]| = |[1,1,1],[x^(2),y^(2),z^(2)],[x^(3),y^(3),z^(3)]|

If D_1=|[1, 1, 1],[x^2,y^2,z^2],[x, y, z]| and D_2=|[1, 1, 1],[yz, zx,xy], [x, y, z]| without expanding prove that D_1=D_2

Prove that |[yz-x^2,zx-y^2,xy-z^2],[zx-y^2,xy-z^2,yz-x^2],[xy-z^2,yz-x^2,zx-y^2]| is divisible by (x+y+z), and hence find the quotient.

Using properties of determinant prove 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)

The determinant |[ C(x,1) ,C(x,2), C(x,3)] , [C(y,1) ,C(y,2), C(y,3)] , [C(z,1) ,C(z,2), C(z,3)]|= (i) 1/3xyz(x+y)(y+z)(z+x) (ii) 1/4xyz(x+y-z)(y+z-x) (iii) 1/12xyz(x-y)(y-z)(z-x) (iv) none

The value of |{:(x,x^2-yz,1),(y,y^2-zx,1),(z,z^2-xy,1):}| is

Using properties of determinats prove that : |(x,x(x^(2)+1),x+1),(y,y(y^(2)+1),y+1),(z,z(z^(2)+1),z +1)|=(x-y)(y-z)(z-x)(x+y+z)

If x, y, z are different and Delta=|[x, x^2, 1+x^3],[y, y^2, 1+y^3],[z, z^2, 1+z^3]|=0 then show that 1+xyz=0