I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative).

The resolutions mentioned in 4th and 5th paragraphs of this question perfectly fit for my purposes but I can't see why these are actually resolutions (that they do not have homology in non-zero grading) and how one can generalize them to higher numbers of variables.

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative).

The resolutions mentioned in 4th and 5th paragraphs of this question perfectly fit for my purposes but I can't see why these are actually resolutions (that they do not have homology in non-zero grading) and how one can generalize them to higher numbers of variables.