If a polynomial equation is solved for multiple roots in a system of equilibria (e.g. calculating extent of a reaction, or solving for [H^{+}] in a complicated acid-base system, to give two basic examples) how do we know which, among the roots, is the correct solution (e.g. the correct extent or the correct proton concentration for the two cases above)? Is it always (in analytical chemistry/equilibrium situations) the smallest real, positive root which is the one we should take? We can assume the root which correctly represents the solution must be real and positive (and often, smaller than a certain upper limit we can impose, as e.g. in the case of extent) but must it be the smallest, and if not, how do we choose the correct root?
So far I have never seen a system form which there will be more than one solution having physical sense.
Neither extent, nor when solving for a concentration? Well you would know... it's surprising that this would be the case even for complicated systems, 8,9,10 degree polynomials etc. So in your experience does the "smallest real positive root" 'rule' always work, or when will it not be physically acceptable? Of course in extent there will only be an upper limit (=number of moles of limiting reagent / stoichiometric coefficient of reagent), lower limit is 0. any situation where this rule doesn't work, off the top of your head?
In all these cases you are solving a system of equations for various concentrations, not just one. All these concentrations are restricted to be positive and eventually there are other restrictions, too. If you take them all into account you will select the correct root of the equation.