a special $n$-simplex, named a prekite, that was introduced and studied in [@prekites], and calculated in Theorem 4.1 there. If we put $a=0$ in $g$, then the instances $t=0$, $t=n$, $t=n-1$, and $t=n-2$ are very closely related to the Cayley-Menger determinants, with respect to certain parametrizations, of orthocentric, isodynamic, circumscriptible, and tetra-isogonic $n$-simplices; see Theorem 5.2 of [@impurity]. Thus the results of this paper would help give complete factorizations of the Cayley-Menger determinants of the aforementioned five special families of $n$-simplices. This raises the question whether the Cayley-Menger determinant $M$ of a general $n$-simplex is irreducible. It is proved in Theorem \[CM\] that the answer is affirmative except in the single case $n=2$, in which case $M$ is the Heron polynomial given in (\[uuu\]). This result has been obtained earlier in [@Siberian] when $k$ is ${\mathbb{R}}$ or ${\mathbb{C}}$. We should note, however, that the result in [@Siberian] does not imply the results in this paper, and does not give any information about the irreducibility of the Cayley-Menger determinant of any of the five special families mentioned earlier. This is because the Cayley-Menger determinants of these special families are obtained from the general Cayley-Menger determinant $M$ given in (\[CMCM\]) by replacing 