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Locally repairable codes (LRCs) are a class of erasure codes that are widelyused in distributed storage systems, which allow for efficient recovery of datain the case of node failures or data loss. In 2014, Tamo and Barg introducedReed-Solomon-like (RS-like) Singleton-optimal $(r,\delta)$-LRCs based onpolynomial evaluation. These constructions rely on the existence of so-calledgood polynomial that is constant on each of some pairwise disjoint subsets of$\mathbb{F}_q$. In this paper, we extend the aforementioned constructions ofRS-like LRCs and proposed new constructions of $(r,\delta)$-LRCs whose codelength can be larger. These new $(r,\delta)$-LRCs are all distance-optimal,namely, they attain an upper bound on the minimum distance, that will beestablished in this paper. This bound is sharper than the Singleton-type boundin some cases owing to the extra conditions, it coincides with theSingleton-type bound for certain cases. Combing these constructions with knownexplicit good polynomials of special forms, we can get various explicitSingleton-optimal $(r,\delta)$-LRCs with new parameters, whose code lengths areall larger than that constructed by the RS-like $(r,\delta)$-LRCs introduced byTamo and Barg. Note that the code length of classical RS codes and RS-like LRCsare both bounded by the field size. We explicitly construct theSingleton-optimal $(r,\delta)$-LRCs with length $n=q-1+\delta$ for any positiveintegers $r,\delta\geq 2$ and $(r+\delta-1)\mid (q-1)$. When $\delta$ isproportional to $q$, it is asymptotically longer than that constructed viaelliptic curves whose length is at most $q+2\sqrt{q}$. Besides, it allows moreflexibility on the values of $r$ and $\delta$.

New constructions of optimal $(r,\delta)$-LRCs via good polynomials