  It is customary to apply the so-called narrow width approximation
$\Gamma(B\to RP_3\to P_1P_2P_3)=\Gamma(B\to RP_3)Br(R\to P_1P_2)$ to extract
the branching fraction of the quasi-two-body decay $B\to RP_3$, with $R$ and
$P_3$ being an intermediate resonant state and a pseudoscalar meson,
respectively. However, the above factorization is valid only in the zero width
limit. We consider a correction parameter $\eta_R$ from finite width effects.
Our main results are: (i) We present a general framework for computing $\eta_R$
and show that it can be expressed in terms of the normalized differential rate
and determined by its value at the resonance. (ii) We introduce a form factor
$F(s_{12},m_R)$ for the strong coupling involved in the $R(m_{12})\to P_1P_2$
decay when $m_{12}$ is away from $m_R$. We find that off-shell effects are
small in vector meson productions, but prominent in the $K_2^*(1430)$,
$\sigma/f_0(500)$ and $K_0^*(1430)$ resonances. (iii) We evaluate $\eta_R$ in
the theoretical framework of QCD factorization (QCDF) and in the experimental
parameterization (EXPP) for three-body decay amplitudes. In general,
$\eta_R^{\rm QCDF}$ and $\eta_R^{\rm EXPP}$ are similar for vector mesons, but
different for tensor and scalar resonances. A study of the differential rates
enables us to understand the origin of their differences. (iv) Finite-width
corrections to $Br(B^-\to RP)_{\rm NWA}$ obtained in the narrow width
approximation are generally small, less than 10\%, but they are prominent in
$B^-\to\sigma/f_0(500)\pi^-$ and $B^-\to \overline K_0^{*0}(1430)\pi^-$ decays.
The EXPP of the normalized differential rates should be contrasted with the
theoretical predictions from QCDF calculation as the latter properly takes into
account the energy dependence in weak decay amplitudes.
