[" Discuss the nature of the roots of the cubic: "],[" (i) "a_(0)x^(3)+3a_(1)x^(2)+3a_(2)x+a_(3)=0]

Let alpha,beta and gamma be the roots of the cubic equation a_(0)x^(3)+3a_(1)x^(2)+3a_(2)x+a_(3)=0(a_(0)!=0) Then teh values of (alpha-beta)^(2)+(beta-gamma)^(2)+(gamma-alpha)^(2)=

Discuss the nature of roots of -x^(2) + 3x + 1 = 0

If a_(1),a_(2),a_(3),a_(4) are are the roots of the equation 3x^(4)-(1+m)x^(3)+2x+5l=0 and Sigma a_(1)=3,a_(1),a_(2),a_(3),a_(4),=10 then (1,m)=

The number of real roots of the equation 1+a_(1)x+a_(2)x^(2)+………..a_(n)x^(n)=0 , where |x| lt (1)/(3) and |a_(n)| lt 2 , is

The number of real roots of the equation 1+a_(1)x+a_(2)x^(2)+………..a_(n)x^(n)=0 , where |x| lt (1)/(3) and |a_(n)| lt 2 , is

If a_(0),a_(1),a_(2),a_(3) are all the positive,then 4a_(0)x^(3)+3a_(1)x^(2)+2a_(2)x+a_(3)=0 has least one root in (-1,0) if ( a )a_(0)+a_(2)=a_(1)+a_(3) and 4a_(0)+2a_(2)>3a_(1)+a_(3)(b)4a_(0)+2a_(2)<3a_(1)+a_(3)(c)4a_(0)+2a_(2)=3a_(1)+a_(0) and 4a_(0)+a_(2)

If a_(0),a_(1),a_(2),.... be the coefficients in the expansion of (1+x+x^(2))^(n) in ascending powers of x ,prove that : (i)a_(0)a_(1)-a_(1)a_(2)+a_(2)a_(3)-...=0