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)`.

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+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 : |(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 the properties of determinant, prove that |(a^(2) +1, ab, ac),(ab, b^(2) + 1, bc),(ac, bc, c^(2)+1)| = 1+a^(2) + b^(2) + c^(2) .

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)|

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)|

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

Prove that (a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)