If `S_r = alpha^r + beta^r + gamma^r` then show that `|[S_0,S_1,S_2],[S_1,S_2,S_3],[S_2,S_3,S_4]| = (alpha-beta)^2(beta-gamma)^2(gamma-alpha)^2`

If S_(r)=alpha^(r)+beta^(r)+gamma^(r) then show that det[[S_(0),S_(1),S_(2)S_(1),S_(2),S_(3)S_(2),S_(3),S_(4)]]=(alpha-beta)^(2)(beta-gamma)^(2)(gamma-alpha)^(2)det[[S_(1),S_(2),S_(3)S_(2),S_(3),S_(4)]]=(alpha-beta)^(2)(beta-gamma)^(2)(gamma-alpha)^(2)

If S_(r)=alpha^(r)+beta^(r)+gamma^(r) then show that det[[S_(2),S_(1),S_(2)S_(1),S_(2),S_(3)S_(2),S_(3),S_(4)]]=(alpha-beta)^(2)(beta-gamma)^(2)(gamma-alpha)^(2)det[[S_(1),S_(2),S_(3)S_(2),S_(3),S_(4)]]=(alpha-beta)^(2)(beta-gamma)^(2)(gamma-alpha)^(2)

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

Show that | (1,1,1), (alpha ^ 2, beta ^ 2, gamma ^ 2), (alpha ^ 3, beta ^ 3, gamma ^ 3) | = (alpha-beta) (beta-gamma) (gamma-alpha) (alphabeta + betagamma + gammaalpha) |

If alpha,beta,gamma are the zeros of the polynomial f(x)=x^(3)-px^(2)+qx-r, then (1)/(alpha beta)+(1)/(beta gamma)+(1)/(gamma alpha)=

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)

Suppose alpha, beta, gamma are roots of x^(3) + qx + r = 0 . If (alpha - beta)^(2) + (beta - gamma)^(2) + (gamma - alpha)^(2) = - e, then q = ________

If alpha, beta, gamma are the roots of x^3+px^2+qx+r=0 then alpha^(2)(beta+gamma)+beta^(2)(gamma+alpha)+gamma^(2)(alpha+beta)=