Why only one secret value with Shamir's secret sharing?
What you're looking for is called packed secret sharing. It was introduced by Franklin &amp; Yung in:


Franklin, M. and Yung, M., Communication complexity of secure computation.. STOC 1992.


If you have a polynomial of degree $&lt; d$, with at most $t$ corrupt parties, then you can use a single polynomial to hide $d - t$ secrets. It's not hard to see that you can't hide more secrets than this. To share $k$ secrets, you need $k$ degrees of freedom. The $d$ coefficients induce $d$ degrees of freedom, but each of the $t$ corrupt parties learns one linear constraint.

Note: in what you described in your post you have a polynomial of degree $&lt; 4$ so if you take the standard Shamir setting of $t=3$ corrupt parties you can only hide the one secret value. While it is true that 3 shares together don't reveal any individual secret coefficient, they reveal linear relationships that constrain the secret coefficients to a 1-dimensional subspace.