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