"Resultant" of three polynomials
Solution 1:
This can be done using resolvants - which date back to Kronecker if not earlier. Below is an elementary introduction from p.14 of Elkadi et al. Algebraic geometry and geometric modeling.
Solution 2:
There is a resultant for three polynomials in two variables. It's considerably trickier than the resultant for two polynomials in one variable.