Legendre polynomials roots and the F-pure threshold of bivariate forms

We provide a direct computation of the $F$-pure threshold degree four homogeneous polynomial in two variables and, more generally, certain polynomials with distinct roots. The depends on whether cross ratio roots satisfies specific M\"{o}bius transformation Legendre polynomial. then make connection between long lasting open question, involving relationship and log canonical threshold, over $\mathbb{F}_p$.