Show that `a-b` is a factor of `|[1,a,a^(2)],[1,b,b^(2)],[1,c,c^(2)]|`

Show that |[1,a,a^2],[1,b,b^2],[1,c,c^2]|=(a-b)(b-c)(c-a)

Show without expanding that |[1,a, a^2],[ 1,b,b^2],[ 1,c,c^2]|=|[1,b c, b+c],[1,c a, c+a],[1,a b, a+b]|

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 ]|

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]|

Using factor theorem,show that a-b,b-C and C-a are the factors of a(b^(2)-c^(2))+b(c^(2)-a^(2))+c(a^(2)-b^(2))

Prove: |(1,a, b c),(1,b ,c a),(1,c ,a b)|=|(1,a ,a^2),( 1,b,b^2),( 1,c,c^2)|

Prove that |(1,a,a^2),(1,b,b^2),(1,c,c^2)|=(a-b)(b-c)(c-a)

If |[a, b, c], [a^(2), b^(2), c^(2)], [a^(3)+1, b^(3)+1, c^(2)+1]|=0 and the vectors given by A(1, a, a^(2)), B(1, b, b^(2)), C(1, c, c^(2)) are non-collinear, then abc=

If [[a,a^(2),a^(3)-1b,b^(2),b^(3)-1c,c^(2),c^(3)-1]]=0

Using factor theorem,show that (a-b),(b-c) and (c-a) are the factors of a(b^(2)-c^(2))+b(c^(2)-a^(2))+c(a^(2)-b^(2))