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

If alpha, beta, gamma are the roots of the equation x^3 + px^2 + qx + r = n then the value of (alpha - 1/(beta gamma)) (beta -1/(gamma alpha)) (gamma-1/(alpha beta)) is:

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 x^3+a x^2+b=0 then the value of [[alpha , beta , gamma],[beta , gamma , alpha],[gamma , alpha , beta]] 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 alpha,beta,gamma are the roots of 4x^3-6x^2+7x+3=0 then find the value of alpha beta +beta gamma+gamma alpha .

If alpha,beta,gamma are the roots of 4x^3-6x^2+7x+3=0 then find the value of alpha beta +beta gamma+gamma alpha .

If alpha, beta, gamma are the roots of the equation x^(3)+4x+1=0 , then find the value of (alpha+beta)^(-1) +(beta +gamma)^(-1) + (gamma+ alpha)^(-1)