If `alpha,betaandgamma` are the cube roots of `P(p)lt0),` then for any `x, y,andz,(xalpha+ybeta+zgamma)/(xbeta+ygamma+zalpha` is equal to

If alpha,beta,gamma are the cube roots of p, then for any x,y,z (x alpha + y beta + z gamma)/(x beta + y gamma + z alpha =

If alpha , beta , gamma are cube roots of p lt 0 , then for any x , y,z (alpha^(2)x^(2)+beta^(2)y^(2)+gamma^(2)z^(2))/(beta^(2)x^(2) + gamma^(2)y^(2) + alpha^(2)z^(2)) is

If alpha , beta are the roots of x^2-p(x+1)+c=0 then (1+alpha )( 1+ beta )=

If alpha,beta are roots of x^2+p x+1=0a n dgamma,delta are the roots of x^2+q x+1=0 , then prove that q^2-p^2=(alpha-gamma)(beta-gamma)(alpha+delta)(beta+delta) .

If alpha, beta are the roots of the equation x^(2)+alphax + beta = 0 such that alpha ne beta and ||x-beta|-alpha|| lt alpha , then

If alpha, beta are the roots of x^(2) - px + q = 0 and alpha', beta' are the roots of x^(2) - p' x + q' = 0 , then the value of (alpha - alpha')^(2) + (beta + alpha')^(2) + (alpha - beta')^(2) + (beta - beta')^(2) is

If alpha,beta are the roots of x^2+p x+q=0 and gamma,delta are the roots of x^2+p x+r=0, then ((alpha-gamma)(alpha-delta))/((beta-gamma)(beta-delta))= (a) \ 1 (b) \ q (c) \ r (d) \ q+r

Let p(x) =x^6-x^5-x^3-x^2-x and alpha, beta, gamma, delta are the roots of the equation x^4-x^3-x^2-1=0 then P(alpha)+P(beta)+P(gamma)+P(delta)=

If tan alpha tan beta are the roots of the equation x^2 + px +q =0(p!=0) then

If alpha,beta are the roots of x^2+p x+q=0a n dgamma,delta are the roots of x^2+r x+s=0, evaluate (alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta) in terms of p ,q ,r ,a n dsdot Deduce the condition that the equation has a common root.