Suppose implement the Shamir secret sharing as following: we select a degree $d$ polynomial $P$ with a zero coefficient of 0, and all other coefficents selected randomly from $Z_p$; and to this polynomial $P$, we add the secret as a constant term $Q = P + secret$. If we use $Q$ as the polynomial in our Secret Sharing scheme, this is precisely equivalent to Shamir's scheme.
Now my question is : what if we multiply instead of add? That is, what if we select $P$ as above, except with a random 0th coefficient as well, and compute $Q = P \times secret$? This makes the original polynomial recover harder, but my main concern is about the (semantically) security of the scheme.