For a field E of characteristic different from 2 and cohomological 2-dimension one, quadratic forms over the rational function field E(X) are studied. A characterisation in terms of polynomials in E[X] is obtained for having that quadratic forms over E(X) satisfy a local-global principle with respect to discrete valuations that are trivial on E. In this way new elementary proofs for the local-global principle are achieved in the cases where E is finite or pseudo-algebraically closed. The study is complemented by various examples.