let `|{:(1+x,x,x^(2)),(x,1+x,x^(2)),(x^(2),x,1+x):}|=(1)/(6)(x-alpha_(1))(x-alpha_(2))(x-alpha_(3))(x-alpha_(4))` be an identity in x, where `alpha_(1),alpha_(2),alpha_(3),alpha_(4)` are independent of x. Then find the value of `alpha_(1)alpha_(2)alpha_(3)alpha_(4)`

if sqrt(alpha_(1)-1)+2sqrt(alpha_(2)-4)+3sqrt(alpha_(3)-9)+4sqrt(alpha_(4)-16)=(alpha_(1)+alpha_(2)+alpha_(3)+alpha_(4))/(2) where alpha_(1),alpha_(2),alpha_(3),alpha_(4) are all real. Then

If alpha and beta are zeroes of 8x^(2)-6x+1 , then find the value of (1)/(alpha)+(1)/(beta) .

If alpha and beta are zeroes of 5x^(2)-7x+1 , then find the value of (1)/(alpha)+(1)/(beta) .

If alpha_(1),alpha_(2),alpha_(3), alpha_(4) are the roos of x^(4)+2x^(3)+bx^(2)+cx+d=0 such that alpha_(1)-alpha_(3)=alpha_(4)-alpha_(2) , then b-c is equal to__________

If alpha_1, ,alpha_2, ,alpha_n are the roots of equation x^n+n a x-b=0, show that (alpha_(1)-alpha_(2))(alpha_(1)-alpha_(3))...(alpha_(1)-alpha_(n))=nalpha_(1)^(n-1)+nalpha

The variance of observation x_(1), x_(2),x_(3),…,x_(n) is sigma^(2) then the variance of alpha x_(1), alpha x_(2), alpha x_(3),….,alpha x_(n), (alpha != 0) is

If alpha,beta are the roots of the equation x^(2)+x+1=0 , find the value of alpha^(3)-beta^(3) .

If the roots of equation x^(3) + ax^(2) + b = 0 are alpha _(1), alpha_(2), and alpha_(3) (a , b ne 0) . Then find the equation whose roots are (alpha_(1)alpha_(2)+alpha_(2)alpha_(3))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(2)alpha_(3)+alpha_(3)alpha_(1))/(alpha_(1)alpha_(2)alpha_(3)), (alpha_(1)alpha_(3)+alpha_(1)alpha_(2))/(alpha_(1)alpha_(2)alpha_(3)) .

If alpha and beta are the roots of 2x^(2) + 5x - 4 = 0 then find the value of (alpha)/(beta) + (beta)/(alpha) .

If 1,1,alpha are the roots of x^3 -6x^2 +9 x-4=0 then find ' alpha'