Unbalance and bow are found to be one of the most common causes of synchronous machinery vibrations in rotating systems. Concentrated lumped mass models are adopted in most of the finite element approach for modeling unbalances and subsequent balancing in rotating systems. But this assumption may not be appropriate for long slender rotors with unbalances distributed along the length of the rotor. A polynomial curve for eccentricity distribution with finite element modeling is used to identify the distributed unbalance. The unbalance eccentricity distributions are estimated using the measured vibration responses at a speed below the balancing speed. Modal correction mass required to balance a rotor at its first bending critical speed, having both distributed unbalance and bow is computed knowing the amplification factor at critical speed. The rotor is balanced at its first bending critical speed using modal balancing method in a single trial run and using a single balancing plane. The method thus avoids multiple trial runs required for modal balancing of flexible rotors. This method is verified on an experimental rotor having both bow and unbalance. The concept of quantifying the distributed unbalance using 'Norm' of eccentricity polynomial function is also introduced for the first time. © 2013 Published by Elsevier Ltd.