  Primordial black holes (PBHs) generated in the early Universe are considered
as one of the candidates for dark matter. To produce PBHs with sufficient
abundance, the primordial scalar power spectrum needs to be enhanced to the
order of 0.01. Considering the third-order polynomial potential with polynomial
$\alpha$ attractors, we show that PBHs with the mass about $10^{17}$g can be
produced while satisfying the constraints from the cosmic microwave background
observations at the 2$\sigma$ confidence level. The mass of PBHs produced in
the polynomial $\alpha$ attractors can be much bigger than that in the
exponential $\alpha$ attractors. By adding a negative power-law term to the
polynomials, abundant PBHs with different masses and the accompanying
scalar-induced gravitational waves (SIGWs) with different peak frequency are
easily generated. The PBHs with masses around $10^{-15}-10^{-12}$ $M_\odot$ can
account for almost all dark matter. The SIGWs generated in the nanohertz band
can explain the recent detection of stochastic gravitational-wave background by
the pulsar timing array observations. The non-Gaussianity of the primordial
curvature perturbations in the squeezed and equilateral limits are calculated
numerically. We find that the non-Gaussianity correction greatly enhances the
PBH abundance which makes the production of PBHs much easier, but the effect of
non-Gaussianity on the generation of SIGWs is negligible.
