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The $K$-theory of a polynomial ring $R[t]$ contains the $K$-theory of $R$ asa summand. For $R$ commutative and containing $\Q$, we describe$K_*(R[t])/K_*(R)$ in terms of Hochschild homology and the cohomology ofK\"ahler differentials for the $cdh$ topology. We use this to address Bass'question, on whether $K_n(R)=K_n(R[t])$ implies $K_n(R)=K_n(R[t_1,t_2])$. Theanswer is positive over fields of infinite transcendence degree; the companionpaper arXiv:1004.3829 provides a counterexample over a number field.

Bass' $NK$ groups and $cdh$-fibrant Hochschild homology