A polymer having a hydrophobic polymer chain derived from monomers of farnesene and other optional monomers, such as dienes and vinyl aromatics. The polymer also includes one or more terminal functional groups, such as an amino group, a glycidyl group, a carboxylic acid group, a (meth)acrylate group, a silane group, an isocyanate group, an acetoacetate group, a phenolic group, and a hydroxyl group. Functional groups, such as carboxylic acids, may also be grafted along the hydrophobic polymer chain. The polymer may be incorporated in curable compositions that optionally include one or more polymer resins having similar functional groups. Methods for preparing the curable polymer compositions are also provided. The curable or cured form of the polymer composition may be used in various products, such as a sealant, a coating, a caulk, an electric potting compound, a membrane, a sponge, a foam, an adhesive, or a propellant binder.