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.