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

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

Show that |{:(1,a,a^2-bc),(1,b,b^2-ca),(1,c,c^2-ab):}|=0

Using determinant : show that the area of the triangle with vertices at (a^2,a^3),(b^2,b^3)and (c^2,c^3) is 1/2(a-b)(b-c)(c-a)(ab+bc+ca) square unit.

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

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

|(1,a^(2)+bc,a^(3)),(1,b^(2)+ac,b^(3)),(1,c^(2)+ab,c^(3))|=-(a-b)(b-c)(c-a)(a^(2)+b^(2)+c^(2))

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) .

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

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)|