There are general solutions for 3rd and 4th degree equations, but it is highly doubtful that you would want to use them in a practical application. Unless it's quadritic or linear, finding the interest rate is a job for numerical methods.
Did you ever notice that your calculator solves immediately for Term, Present Value, Future Value, or Payment Amount, but takes sometimes quite a while to find a missing interest Rate? This is why. Direct solutions are available for everything except the interest rate.