Another way to write the Hochschild homology is as follows:
take A as a bimodule over itself, take a free resolution as a bimodule, and then apply the functor of coinvariants (M -> M/(rm-mr)).
Your definition used the "bar-complex" resolution of the form --> A (x) A (x) A --> A (x) A but k[t] has a much nicer resolution as a bimodule over itself, the Koszul resolution.
This is of the form k[t] (x) k[t] -> k[t] (x) k[t] with the map given by 1 (x) t - t (x) 1, so when you apply coinvariants, you get two copies of k[t] with trivial differential.
Actually all Koszul algebras have a nice resolution of the diagonal bimodule, and thus its easier to compute their Hochschild homology, though in general, they don't always have trivial differential after applying coinvariants.