Prove that `|[a,b,c] , [a^2,b^2,c^2] , [a^3,b^3,c^3]|= abc(a-b)(b-c)(c-a)`

Show that |[a,b,c],[a^2,b^2,c^2],[a^3,b^3,c^3]|=abc(a-b)(b-c)(c-a)

Show that |[b+c,c+a,a+b] , [a+b,b+c,c+a] , [a,b,c]|=a^3+b^3+c^3-3abc

The value of the determinant |(1,a,a^2-bc),(1,b,b^2-ca),(1,c,c^2-ab)| is (A) (a+b+c),(a^2+b^2+c^2) (B) a^3+b^3+c^3-3abc (C) (a-b)(b-c)(c-a) (D) 0

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

Prove that: |[a-b-c, 2a,2a],[2b,b-c-a,2b],[2c,2c,c-a-b]|=(a+b+c)^3

Prove that |[(a+b)^(2),ca,bc],[ca,(b+c)^(2),ab],[bc,ab,(c+a)^(2)]|=2abc(a+b+c)^(3)

Prove that |(a+b, b, c), (b+c, c, a), (c+a, a, b)| = 3abc - a^3-b^3-c^3.

Prove: |a^3 2a b^3 2b c^3 2c|=2(a-b)(b-c)(c-a)(a+b+c)

Using properties of determinants,prove that: (a+b)^(2),ca,cbca,(c+b)^(2),abcb,ab,(c+a)^(2)]]=2abc(a+b+c)^(3)

Prove that : |{:(1,b,c),(b,c,a),(c,a,b):}|=3 a b c-a^(3)-b^(3)-c^(3)