Find polynomial whose root is sum of roots of other polynomials We have two numbers $\alpha$ and $\beta$.
We know that $\alpha$ is root of polynomial $P_n(x)$ of degree $n$ and $\beta$ is root of polynomial $Q_m(x)$ of degree $m$.
How do you find polynomial $R_{n m}(x)$ which has root equal to $\alpha+\beta$ without finding values of roots?
All polynomials are with integer coefficients.
One more question, can it be found using matrix determinant?

If you have two polynomials, say $P(x)$ and $Q(y)$ then, their Resultant, $\operatorname{Res}(P,Q)$ is the determinant of their Sylvester matrix. It equals to
$$
\operatorname{Res}(P,Q)=\prod_{P(\alpha)=0, \ P(\beta)=0,}(\alpha-\beta),
$$
the product of all the differences of their roots.
Now if you consider the two variable polynomial $Q(z-y)$ as a polynomial in the variable $y$ (and therefore its roots are $\{z-\beta: \ Q(\beta)=0\}$) then, the resultant $\operatorname{Res}(P,Q)$ is a polynomial in the variable $z$ with roots $\alpha+\beta$ where $\alpha$ runs over the roots of $P$ and $\beta$ runs over the roots of $Q$.
