I need to solve $p(x)=aq(x)$ with multiple real $a$, where $p(x)$ and $q(x)$ are the two polynomials in $x$ (with real coefficients). The roots of $p(x)$ and $q(x)$ were found previously, i.e. these two polynomials are factored.
Is it somehow possible to solve the equation in question without multiplying $q(x)$ by $a$ and transferring everything to the left for every new $a$? Is there any workaround?
It's quite frustrating that the two polynomials $p(x)$ and $q(x)$ are known (together with their roots), but every time I substitute a new parameter value ($a$ in this case) I need to solve a totally new polynomial.
Sorry if the question is stupid. Actually I don't really believe that there's any workaround, although finding one would be really nice.
The problem is solved numerically, all polynomial roots are found as eigenvalues of the polynomial's companion matrix.