The standard polynomial resolution has an augmentation contraction and is admissible

The extra degeneracy. Let $t\text{\hspace{0.17em}⁣}:P_n\to P_{n+1}$ be the map $x\mapsto [x]$ induced by the unit of the free-forgetful adjunction, including the augmented map $B\to P_0=A[B]$. Directly on the nested polynomial expressions defining $P$, the adjunction triangle identities give $d_{0}t=\mathrm{i}\mathrm{d}$, $d_{i+1}t=t{d}_i$, $s_{i+1}t=t{s}_i$ and $s_{0}t=tt$, with the augmented interpretations in degrees $-1$ and $0$. These identities say that $t$ is an extra degeneracy for the augmented simplicial set underlying $P_{\bullet }\to B$.

Homotopy over $B$. For an order-preserving map $\alpha \text{\hspace{0.17em}⁣}:[n]\to [1]$ with $r$ initial zeros, consider the map $t^r{d}_0^r\text{\hspace{0.17em}⁣}:P_n\to P_n$, where $d_0^r$ is interpreted using the augmentation when $r=n+1$. Write this map as $H_{n,r}$. The identities of step 1.1 give $d_j{H}_{n,r}=H_{n-1,r-1}{d}_j$ for $j<r$ and $d_j{H}_{n,r}=H_{n-1,r}{d}_j$ for $j\ge r$; similarly $s_j{H}_{n,r}=H_{n+1,r+1}{s}_j$ for $j<r$ and $s_j{H}_{n,r}=H_{n+1,r}{s}_j$ for $j\ge r$. At the all-zero endpoint $r=n+1$, the repeated face map lands in $B$ and the same identities use the augmentation. Deleting or repeating the $j$-th vertex of $\alpha$ changes its number of initial zeros by exactly the stated amount. Since faces and degeneracies generate all order maps, these equations prove simplicial naturality, so the maps assemble into a simplicial homotopy over $B$ from the composite of the augmentation with the constant section $b\mapsto [\dots [b]\dots ]$ (the all-zero endpoint) to the identity of $P_{\bullet }$ (the all-one endpoint). Augmentation followed by that section is therefore homotopic to the identity, while the other composite is the identity on $B$; hence the augmentation is a homotopy equivalence of underlying simplicial sets over $B$. Every augmentation map $P_n\to B$ is surjective because the nested variables $[b]$ lift every $b\in B$.

Admissibility. By [F2] the underlying-set homotopy equivalence of step 2.1 makes the associated chain map of abelian groups a quasi-isomorphism, and the termwise surjectivity of the augmentation then makes $P_{\bullet }\to B$ a trivial Kan fibration. Each $P_n$ is a free $A$-module by [F1], so the associated complex, reindexed cohomologically in nonpositive degrees, is a complex of free $A$-modules with $H^0=B$ and vanishing higher homology; it is therefore a free resolution of $B$, and $P_{\bullet }$ is admissible for computing cotangent complexes. The extra degeneracy is only a map of sets, not an algebra-linear chain contraction; it is the normalization and prism lemma [F2] that passes the contraction from underlying simplicial sets to module homology.