  We generalize the factorized resummation of multipolar waveforms introduced
by Damour, Iyer and Nagar to spinning black holes. For a nonspinning
test-particle spiraling a Kerr black hole in the equatorial plane, we find that
factorized multipolar amplitudes which replace the residual relativistic
amplitude f_{l m} with its l-th root, \rho_{l m} = f_{l m}^{1/l}, agree quite
well with the numerical amplitudes up to the Kerr-spin value q \leq 0.95 for
orbital velocities v \leq 0.4. The numerical amplitudes are computed solving
the Teukolsky equation with a spectral code. The agreement for prograde orbits
and large spin values of the Kerr black hole can be further improved at high
velocities by properly factoring out the lower-order post-Newtonian
contributions in \rho_{l m}. The resummation procedure results in a better and
systematic agreement between numerical and analytical amplitudes (and energy
fluxes) than standard Taylor-expanded post-Newtonian approximants. This is
particularly true for higher-order modes, such as (2,1), (3,3), (3,2), and
(4,4) for which less spin post-Newtonian terms are known. We also extend the
factorized resummation of multipolar amplitudes to generic mass-ratio,
non-precessing, spinning black holes. Lastly, in our study we employ new,
recently computed, higher-order post-Newtonian terms in several subdominant
modes, and compute explicit expressions for the half and one-and-half
post-Newtonian contributions to the odd-parity (current) and even-parity (odd)
multipoles, respectively. Those results can be used to build more accurate
templates for ground-based and space-based gravitational-wave detectors.
