Question

Find the condition which must be satisfied by th coordinates of the polynomial f(x)= x^3-px^2+ax-r when the sum of its two zeros is zero.

Answer 1

Say $\alpha, -\alpha, \beta$ are the roots of given polynomial.
$\alpha - \alpha + \beta = p \implies \beta = p$
That means  $p$ is a zero of the polynomial. Plug this in the polynomial and the required condition would be
$0 = p^3 - p(p)^2 + a(p)-r\\0=ap-r\\ap=r$