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

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)]|

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

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)

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

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 : Det[[x,x^2,x^3],[y,y^2,y^3],[z,z^2,z^3]]=xyz(x-y)(y-z)(z-x)

If Delta_(1)=|{:(Ax,x^(2),1),(By,y^(2),1),(Cz,z^(2),1):}|" and "Delta_(2)=|{:(A,B,C),(x,y,z),(yz,zx,xy):}| , then

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

If |(x^k,x^(k+2),x^(k+3)), (y^k,y^(k+2),y^(k+3)), (z^k,z^(k+2),z^(k+3))|=(x-y)(y-z)(z-x){1/x+1/y+1/z} then k=