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

Prove that |(-a^(2),ab,ac),(ab,-b^(2),bc),(ac,bc,-c^(2))| = 4a^(2)b^(2)c^(2) .

Prove that |(a,a+b,a+b+c),(2a,3a+2b,4a+3b+2c),(3a,6a+3b,10a+6b+3c)| = a^(3)

Prove that 8a^3-(a+b)^3-(a-c)^3-(c-b)^3=3(a-b)(a+c)(2a+b-c)

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

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)

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

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