Sometimes we have the same fixed set of polynomial points $\vec x$ and index $j$, and we're computing lagrange coefs $\text{coef}(j, x, \vec x)$ for different $x$. Recall that lagrange coef is computes as:
$$\text{coef}(j, x, \vec x) = \frac{\prod_{m \ne j} x - \vec x_m}{\prod_{m \ne j} \vec x_j - \vec x_m}$$
Note that denominator is independent of $x$. We can compute
$$\text{denom}_{j, \vec x} = (\prod_{m \ne j} \vec x_j - \vec x_m)^{-1}$$
then we can compute lagrange coef without doing modular inversion:
$$\text{coef}'(x, \text{denom}) = \text{denom} \cdot \prod_{m \ne j} x - \vec x_m$$
