Prove that
`|{:(1,x+alpha,y+z-alpha),(1,t+beta,+x-beta),(1,z+gamma,x+y-gamma):}|=0`

Prove that | [1, alpha, alpha ^ 2 + betagamma], [1, beta, beta ^ 2 + gammaalpha], [1, gamma, gamma ^ 2 + alphabeta] | = 2 (alpha-beta) (beta-y ) (y-alpha)

Show that |[1,alpha,alpha^2],[1,beta,beta^2],[1,gamma,gamma^2]|=(alpha-beta)(beta-gamma)(gamma-alpha)

If alpha,beta,gamma are different from 1 and are the roots of ax^(3)+bx^(2)+cx+d=0 and (beta-gamma)(gamma-alpha)(alpha-beta)=(25)/(2) ,then prove that det[[(alpha)/(1-alpha),(beta)/(1-beta),(gamma)/(1-gamma)(alpha)/(1-alpha),(beta)/(1-beta),(gamma)/(1-gamma)alpha,beta,gammaalpha^(2),beta^(2),gamma^(2)]]=(25d)/(2(a+b+c+d))

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

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

.If alpha,beta,gamma are roots of x^(3)-x^(2)-1=0 then the value of ((1+alpha))/((1-alpha))+((1+beta))/((1-beta))+((1+gamma))/((1-gamma)) is equal to

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

Prove that |[alpha,beta,gamma] ,[alpha^2,beta^2,gamma^2] , [beta+gamma, gamma+alpha, beta+alpha]| = (alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)