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

Using the properties of determinants show that : |[[1,1,1],[a^2,b^2,c^2],[a^3,b^3,c^3]]|=(a-b)(b-c)(c-a)(ab+bc+ca) .

Show that |[a ,b ,c],[ a^2,b^2,c^2],[bc, ca, ab]|=|[1, 1, 1],[a^2,b^2,c^2],[a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a b+b c+c a) .

Show that |{:(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3):}| =(a-b)(b-c)(c-a)(ab+bc+ca)

Prove that |[1,1,1] , [a,b,c] ,[a^2-bc, b^2-ca, c^2-ab]|=0

Prove that: 1/(bc+ca+ab)|[a, b, c],[a^2, b^2, c^2], [bc, ca, ab]|=(b-c),(c-a),(a-b)

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

Match the following from List - I to List - II {:("List-I","List-II"),((I)|{:(1,1,1),(a,b,c),(bc,ca,ab):}|=,(a)(a-b)(b-c)(c-a)),((II)|{:(a,b,c),(a^(2),b^(2),c^(2)),(a^(3),b^(3),c^(3)):}|=,(b)(a-b)(b-c)(c-a)abc),((III)|{:(1,1,1),(a,b,c),(a^(3),b^(3),c^(3)):}|=,(c)(a-b)(b-c)(c-a)(a+b+c)):}

Prove that abs((a,b,c),(a^2,b^2,c^2),(bc,ca,ab))=(a-b)(b-c)(c-a)(ab+bc+ca)

Without expanding show that : |(1,a,a^2),(1,b,b^2),(1,c,c^2)|=|(1,bc,b+c),(1,ca,c+a),(1,ab,a+b)|

Show that |abca^(2)b^(2)c^(2)baab|=|111a^(2)b^(2)c^(2)a^(2)b^(2)c^(3)|=(a-b)(b-c)(c-a)(ab+bc+ca)