FAMOUS 3 SQURE PROBLEM | Can You Find The Value of `alpha + beta + gamma` ?

The origin is the centroid of /_\ABC with the vertices A(alpha, 1, 3),B(-2, beta, -5) and C(4,7,gamma) find the values of alpha, beta, gamma .

If alpha, beta, gamma are the roots of the equation x^(3) + x + 1 = 0 , then the value of alpha^(3) + beta^(3) + gamma^(3) , is

Let cos(alpha-beta) + cos(beta - gamma) + cos(gamma - alpha)= -3/2 , then the value of cos alpha + cos beta + cos gamma is

If origin is the centroid of /_\ABC with the vertices A(alpha, 1,3),B(-2,beta, -5) and C(4,7,gamma) find the value of alpha, beta,gamma

If roots of x ^(3)+2x ^(2) +1=0 are alpha, beta and gamma, then the value of (alpha beta )^(3)+ (beta gamma ) ^(3) + (alpha gamma )^(3), is:

If each of alpha beta and gamma is a positive acute angle such that sin(alpha+beta-gamma)=(1)/(2)cos(beta+gamma-alpha)=(1)/(2) and tan(gamma+alpha-beta)=1 find the values of alpha,beta and gamma

If alpha, beta and gamma are the roots of the equation x^(3) + 3x^(2) - 24x + 1 = 0 thebn find the value of (3sqrt(alpha)+3sqrt(beta)+ 3sqrt(gamma)).

If alpha,beta gamma are zeroes of cubic polynomial x^(3)+5x-2 , then find the value of alpha^(3)+beta^(3)+gamma^(3) .

If alpha, beta, gamma are the roots of x^(3)+ax+b=0 , then find the value of alpha^(3)+beta^(3)+gamma^(3) .