The achievable bandwidth of common linear-phase RF pulses is limited by the maximum feasible B1 amplitude of the MR system. It has been shown previously, that this limitation can be circumvented by overlaying a quadratic phase in the frequency domain, which spreads the power across the pulse duration. Quadratic-phase RF pulses are near optimal in terms of achieving minimal B1max. In this work, it is demonstrated that further B1max reduction can be achieved by combining quadratic with higher-order polynomial-phase functions. RF pulses with a phase response up to tenth order were designed using the Shinnar-Le Roux transformation, yielding considerable increases in bandwidth and selectivity as compared to pure quadratic-phase pulses. These benefits are studied for a range of pulse specifications and demonstrated experimentally. For B1max = 20 microT and a pulse duration of 2.1 ms, it was possible to increase the bandwidth from 3.1 kHz for linear and 3.8 kHz for a quadratic to 9.9 kHz for a polynomial-phase pulse.