If `a+b+c=0` then prove that `a^3+b^3+c^3=3 abc`

If a+b+c=5 and ab+bc+ca=10 , then prove that a^3+b^3+c^3-3abc=-25 .

If a+b+c=5 and ab+bc+ca=10, then prove that a^(3)+b^(3)+c^(3)-3abc=-25 .

If a, b, c are in GP, prove that a^ 2 b^ 2 c ^ 2 ( 1/ a ^ 3 +1/ b ^ 3 ​ +1/ c ^ 3 ​ )=a ^ 3 +b^ 3 +c^ 3 .

If a+b+c=0, then write the value of a^3+b^3+c^3

If a + 2b + c= 0 , then show that: a^(3) + 8b^(3) + c^(3)= 6abc

omega is an imaginary root of unity. Prove that (i) ( a + bomega + comega^(2))^(3) + (a+bomega^(2) + comega)^(3) = (2a-b-c)(2b -a -c)(2c -a-b) (ii) If a+b+c = 0 then prove that (a + bomega + comega^(2))^(3)+(a+bomega^(2) + comega)^(3) = 27abc .

omega is an imaginary root of unity. Prove that If a+b+c = 0 then prove that (a + bomega + comega^(2))^(3)+(a+bomega^(2) + comega)^(3) = 27abc .

If a, b, c are in A.P., prove that a^(3)+4b^(3)+c^(3)=3b(a^(2)+c^(2)).

If a+ b + c = 0 , show that : a^(3) + b^(3) + c^(3)= 3abc

If the centroid of the triangle formed by the points (a ,\ b),\ (b ,\ c) and (c ,\ a) is at the origin, then a^3+b^3+c^3= a b c (b) 0 (c) a+b+c (d) 3\ a b c