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 : |{:((y+z)^(2),x^(2),x^(2)),(y^(2),(x+z)^(2),y^(2)),(z^(2),z^(2),(x+y)^(2)):}|=2xyz (x+y+z)^(3)

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

Show that : |[x, y, z ],[x^2,y^2,z^2],[x^3,y^3,z^3]|=x y z(x-y)(y-z)(z-x)dot

prove that: |(y^(2)z^(2),yz,y+z),(z^(2)x^(2),zx,z+x),(x^(2)y^(2),xy,x+y)|=0

Prove that |{:(ax,,by,,cz),(x^(2),,y^(2),,z^(2)),(1,,1,,1):}|=|{:(a,,b,,c),(x,,y,,z),(yz,,xz,,xy):}|

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

In each polynomial, given below, separate the like terms : y^(2)z^(3), xy^(2)z^(3), -5x^(2)yz, -4y^(2)z^(3), -8xz^(3)y^(2), 3x^(2)yz and 2z^(3)y^(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.

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