The roots of the equation `|{:(x,alpha,1),(beta,x,1),(beta,gamma,1):}|=0` are independent of

The root of the equation |(x,alpha,1),(beta,x,1),(beta,gamma,1)| = 0 are independent of :

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

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

If alpha,beta are the roots of the equation x^(2)+px+1=0 gamma,delta the roots of the equation x^(2)+qx+1=0 then (alpha-gamma)(alpha+delta)(beta-gamma)(beta+delta)=

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

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

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

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