Let $p_n=a_0+a_1x+...+a_nx^n$ be a polynomial with only real roots.
I need a way to algorithmically find real coefficients $a_{n+1},a_{n+2},...\neq 0$ such that the polynomial $p_{n+1}=p_n+a_{n+1}X^{n+1}$ has $n+1$ real roots and that they satisfy $$ |r_{n+1}|\geq |r_n| $$
For all roots $r_n, r_{n+1}$ of $p_n, p_{n+1}$, respectively.
Does anyone know an easy way to achieve this?