The most general structure of an elastic partial wave amplitude when the unphysical cuts are neglected is deduced in terms of the N/D method. This result is then matched to lowest order, O(p2), Chiral Perturbation Theory(χPT) and to the exchange (consistent with chiral symmetry) of resonances in the s-channel. The extension of the method to coupled channels is also given. Making use of the former formalism, the ππ and Kπ(I=1/2) P-wave scattering amplitudes are described without free parameters when taking into account relations coming from the 1/Nc expansion and unitarity. Next, the scalar sector is studied and good agreement with experiment up to s√=1.4 GeV is found. It is observed that the a0(980), σ and κ(900) resonances are meson-meson states originating from the unitarization of the O(p2) χPT amplitudes. On the other hand, the f0(980) is a combination of a strong S-wave meson-meson unitarity effect and of a preexisting singlet resonance with a mass around 1 GeV. We have also studied the size of the contributions of the unphysical cuts to the ππ(I=0) and Kπ(I=1/2) elastic S-wave amplitudes from χPT and the exchange of resonances in crossed channels up to s√≈800 MeV. The loops are calculated as in χPT at next to leading order. We find a small correction from the unphysical cuts to our calculated partial waves.