Observation of the decay B-+/- -> pi(+/-)pi(0), study of B-+/- -> K-+/-pi(0), and search for B-0 -> pi(0)pi(0)

We present results for the branching fractions and charge asymmetries in B+/--->h(+/-)pi(0) (where h(+/-)=pi(+/-),K+/-) and a search for the decay B-0-->pi(0)pi(0) using a sample of approximately 88x10(6) B (B) over bar pairs collected by the BABAR detector at the PEP-II asymmetric-energy B Factory at SLAC. We measure B(B+/--->pi(+/-)pi(0))=(5.5(-0.9)(+1.0)+/-0.6)x10(-6), where the first error is statistical and the second is systematic. The B+/--->pi(+/-)pi(0) signal has a significance of 7.7sigma including systematic uncertainties. We simultaneously measure the K(+/-)pi(0) branching fraction to be B(B+/--->K(+/-)pi(0))=(12.8(-1.1)(+1.2)+/-1.0)x10(-6). The charge asymmetries are A(pi +/- pi 0)=-0.03(-0.17)(+0.18)+/-0.02 and A(K +/- pi 0)=-0.09+/-0.09+/-0.01. We place a 90% confidence-level upper limit on the branching fraction B(B-0-->pi(0)pi(0)) of 3.6x10(-6) .