Factorise : `|(a,b,c),(a^2,b^2,c^2),(a^3,b^3,c^3)|`

Using properties of determinant show that : |(a,b,c),(a^2,b^2,c^2),(b+c,c+a,a+b)|=(a+b+c)(a-b)(b-c)(c-a)

Factorise: a^3-a^2b^2

Factorise : |(1,1,1),(b+c,c+a,a+b),(b^2+c^2,c^2+a^2,a^2+b^2)|

Using properties of determinant show that : |(a,b+c,a^2),(b,c+a,b^2),(c,a+b,c^2)|=-(a+b+c)(a-b)(b-c)(c-a)

Without expanding show that : |(bc,a,a^2),(ca,b,b^2),(ab,c,c^2)|=|(1,a^2,a^3),(1,b^2,b^3),(1,c^2,c^3)|

Prove that |(3a, -a+b, -a+c),(a-b, 3b, c-b),(a-c, b-c, 3c)| = 3(a+b+c)(ab+bc+ca) .

Factorise (2a+3b+4c)(6ab+8ac+12bc)-24abc

Factorise (a-2b-3c)(6bc-2ab-3ac)-6abc

If a + b + c = 0 , then prove that (a^2+b^2+c^2)/(a^3+b^3+c^3)+2/3(1/a+1/b+1/c)=0

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