IF `alpha,beta` and `gamma,delta` be the roots of the equation `x^2+px-r=0` and `x^2+px+r=0` respectively, prove that `(alpha-gamma)(alpha-delta)=(beta-gamma)(beta-delta)`

IF alpha,beta and gamma, delta are the roots of the equations x^2-bx+c=0 and x^2-px+q=0 respectively, show that , (alpha-gamma)(beta-delta) -alpha.gamma - beta. delta=(c+q)-bp .

If alpha,beta be the roots x^2+px-q=0 and gamma,delta be the roots of x^2+px+r=0, p+rphi0 ,then ((alpha-gamma)(alpha-delta))/((beta-gamma)(beta-delta)) is equal to

If alpha,beta are the roots x^2+px+q =0 and gamma,delta are the roots of x^2+rx+s =0, evaluate (alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta) in terms of p,q,r and s. Deduce the condition that the equation may have a common root.

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 alpha , beta , gamma are the roots of the equation px^3 - qx + r = 0 , then the value of alpha + beta + gamma is

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,gamma are the roots of the equation x^3+p x^2+q x+r=0, then find he value of (alpha-1/(betagamma))(beta-1/(gammaalpha))(gamma-1/(alphabeta)) .

If alpha,beta,gamma are the roots of the equation x^3-p x+q=0, then find the cubic equation whose roots are alpha/(1+alpha),beta/(1+beta),gamma/(1+gamma) .

If alpha,beta,gamma are the roots of the equation x^3+x+1=0 , then the value of alpha^3+beta^3+gamma^3 is.

If alpha,beta,gamma are the roots of he euation x^3+4x+1=0, then find the value of (alpha+beta)^(-1)+(beta+gamma)^(-1)+(gamma+alpha)^(-1) .