If `p lt 0 and alpha, beta, gamma` are cube roots of p then for any a,b and c show that `(aalpha+b beta + cgamma)/(a beta+bgamma +calpha)=omega^(2)` where `omega` is an imaginary cube root of 1.

If alpha be the real positive cube root and beta,gamma be the complex cube roots of m, a real positive number, then for any x,y,z show that (xbeta+ygamma+zalpha)/(xgamma+yalpha+zbeta)=omega^(2) where omega is a complex cube root of unity.

If alpha, beta, gamma are the roots of x^3 + px + q = 0 , then alpha beta gamma = ___________ .

If alpha , beta , gamma are the roots of the equation px^3 - qx + r = 0 , then the value of alpha + beta + gamma is

|{:(1,omega,omega^2),(omega,omega^2,1),(omega^2,1,omega):}|=0 where omega is an imaginary cube root of unity.

Evaluate: |{:(1,1,1),(1,omega^2,omega),(1,omega,omega^2):}| (where omega is an imaginary cube root of unity ).

Evaluate abs((1,omega,omega^2),(omega^2,1,omega),(omega^2,omega,1)) were omega is an imaginary cube root of unity.

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

If alpha,beta,gaama are the roots of the cubic x^3+qx+r=0 , find the equation whose roots are (alpha-beta)^2,(beta-gamma)^2,(gamma-alpha)^2 .

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

If alpha , beta be the imaginary cube root of unity, then show that alpha^4+beta^4+ alpha^-1beta^-1=0