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)`

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)

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

Using properties of determinant show that : |(a+b+2c,a,b),(c,b+c+2a,b),(c,a,c+a+2b)|=2(a+b+c)^3

Using properties of determinant show that : |(a^2,b^2,c^2),((a+1)^2,(b+1)^2,(c+1)^2),((a-1)^2,(b-1)^2,(c-1)^2)|=4|(a^2,b^2,c^2),(a,b,c),(1,1,1)|

Using properties of determinant show that : |(a^2,bc,c^2+ac),(a^2+ab,b^2,ac),(ab,b^2+bc,c^2)|=4a^2b^2c^2

Using properties of determinants, prove that |(1,a,a^(3)),(1,b,b^(3)),(1,c,c^(3))| = (a-b)(b-c)(c-a)(a+b+c) .

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

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

Without expanding show that : |(a,b+c,1),(b,c+a,1),(c,a+b,1)|=0