`|[1,bc,b+c],[1,ca,c+a],[1,ab,a+b]|=|[1,a,a^2],[1,b,b^2],[1,c,c^2]|`

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

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

1,bc,b+c1,ca,c+a1,ab,a+b]|=det[[1,a,a^(2)1,b,b^(2)1,c,c^(2)]]

|[1,a,bc] , [1,b,ca] , [1,c,ab]|=

|[1,bc,a(b+c)],[1,ca,b(c+a)],[1,ab,c(a+b)]|=0

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]|

det[[bc,b+c,1ca,c+a,1ab,a+b,1]]=

|[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)]|

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