Find polynomial whose root is sum of roots of other polynomials

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 $Q$.

Of course, $x-\alpha-\beta$ has $\alpha+\beta$ as a root, but let's say that $P$ and $Q$ have rational coefficients, and we want $R$ to as well. If the roots of $P$ are $\alpha_1,\ldots,\alpha_n$, and the roots of $Q$ are $\beta_1,\ldots,\beta_n$, then $\prod_{i,j} (x-\alpha_i - \beta_j)$ has rational coefficients (they are fixed by the Galois group) and has $\alpha+\beta$ as a root.

In practice, this approach leads to working with elementary symmetric polynomials, as in the other answer, but I find this view more intuitive.