Evaluate the following: ` |[1,1,1],[alpha, beta, gamma],[beta gamma, gama alpha, alpha beta]| `

Evaluate the following: |[1,1,1],[alpha, beta, gamma],[beta gamma, gamma alpha, alpha beta]|

Without expanding, prove the following |(1,1,1),(alpha,beta,gamma),(betagamma,gammaalpha,alphabeta)|=(beta-alpha)(gamma-alpha)(alpha-beta)

Evaluate: |[alpha, beta,gamma],[alpha^2,beta^2,gamma^2],[beta+gamma,gamma+alpha,alpha+beta]|

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

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)

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

Without expanding the determinant, prove that: |{:(1, beta gamma, beta + gamma),(1, gamma alpha, gamma + alpha),(1, alpha beta, alpha + beta):}| = |{:(1, alpha, alpha^(2)),(1, beta, beta^(2)),(1, gamma, gamma^(2)):}|

Prove that, |{:(alpha,beta,gamma),(alpha^2,beta^2,gamma^2),(beta+gamma,gamma+alpha,alpha+beta):}|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

Using properties of determinant prove that |{:(alpha,beta,gamma),(alpha^2,beta^2,gamma^2),(beta+gamma,gamma+alpha,alpha+beta):}|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

If alpha, beta, gamma are the roots of ax^(3) + bx^(2) + cx + d = 0 and |(alpha,beta,gamma),(beta,gamma,alpha),(gamma,alpha,beta)| = 0, alpha != beta != gamma , then find the equation whose roots are alpha + beta - gamma, beta + gamma - alpha and gamma + alpha - beta