Question

find the collection which must be satisfied by the co-efficient of polynomials , such that the sum of two zeros of the polynomial f(x) =x3 -px2+qx-r is zero

Answer 1

Given polynomial $p(x)=x^3-px^2+qx-r$
Let α,β, $\gamma$ be the roots of p(x)
we know, sum of roots= $\frac{-b}{a}$
$\alpha + \beta + \gamma = p \implies \gamma = p$
& Sum of roots taken two at time = $\frac{c}{a}$
$\alpha \beta +\beta \gamma + \gamma \alpha = q \implies 0=q$
Also Product of roots = $\frac{-d}{a}$
$\alpha \beta \gamma = r \implies 0=r$
∴  for  condition to be satisfied we need q and x to be zero