` |[a ,a^(2), bc ], [b, b^(2), ca],[c ,c^(2), ab] |=| [1 ,a^(2) ,a^(3) ],[1 ,b^(2), b^(3)], [1 ,c^(2), c^(3)] |`

Show without expanding at any stage that: [a,a^2,bc],[b,b^2,ca],[c,c^2,ab]|=|[1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3]|

Without expanding the determinant prove that abs([a,a^2,bc],[b,b^2,ca],[c,c^2,ab])=abs([1,a^2,a^3],[1,b^2,b^3],[1,c^2,c^3])

|[(1)/(a),a^(2),bc],[(1)/(b),b^(2),ca],[(1)/(c),c^(2),ab]|=0

(1)/(a),a^(2),bc(1)/(b),b^(2),ca(1)/(c),c^(2),ab]|

Prove that det[[1,a,a^(2)-bc1,b,b^(2)-ca1,c,c^(2)-ab]]=0

|(1/a,a^2,bc),(1/b,b^2,ca),(1/c,c^2,ab)|=........

det[[a,a^(2),bcb,b^(2),cac,c^(2),ab]]=(a-b)(b-c)(c-a)(ab+bc+ca)

Prove that |[1,a,bc] , [1,b,ca], [1,c,ab]|=|[1,a,a^2] , [1,b,b^2] , [1,c,c^2]|

|[bc,ca,ab],[(b+c)^(2),(c+a)^(2),(a+b)^(2)],[a^(2),b^(2)c^(2)]|