Polymer flooding is a proven enhanced oil recovery method that can significantly improve sweep and oil recovery. Predictions of polymer floods are typically made using numerical simulators, which may or may not include non-Newtonian effects. While simulation is very important, the results of simulations must be tested against known analytical solutions. Fractional-flow theory using method of characteristics is an effective method to calibrate and test numerical simulations. Current research using analytical solutions developed by fractional flow theory are restricted to Newtonian rheology except in rectilinear flow. In this research, we extend the fractional flow theory to a more realistic case of non-Newtonian rheology for a 1-D radial system. In particular, we extend the fractional flow theory to polymer flooding where the non-Newtonian rheology results in a system where the fractional flow curve is position dependent and shifts with increasing radial distance. We present a semi-analytical solution for the changing fractional flow curve system where shocks collide. In addition to the chemical shock and the Buckely-Leverett shock, there is an intermediate fast shock that is formed that determines the water breakthrough time. We explain the resolution of shocks and track the movement of different shocks with increasing radial distances using a semi-analytical approach. The tracking of shocks improves the accuracy of the prediction of polymer and water breakthrough times and hence oil recoveries. The proposed semi-analytical solution is also compared with the finite difference simulation and shows good agreement. In addition to the calibration of simulators, the extended fractional-flow theory for non-Newtonian rheology also provides useful insights for scale-up of laboratory experiments. The analytical solutions, by themselves, can also be input to streamline simulations. These analytical solutions are critical to establish confidence on oil recovery predictions as we continually seek higher conversion of resources to reserves.