1. The problem is to find the shortest distance from the point $P(1,2,1)$ to the sphere \(x^2 + y^2 + z^2 = 3\). 2. First, calculate the distance from the center of the sphere $O(0,0,0)$ to the point $P$. $$d = \sqrt{(1-0)^2 + (2-0)^2 + (1-0)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}$$ 3. The sphere has radius $r = \sqrt{3}$. 4. The shortest distance from the point $P$ to the sphere surface is the distance from $P$ to $O$ minus the radius $r$, given by: $$D = d - r = \sqrt{6} - \sqrt{3}$$ 5. Since both $\sqrt{6}$ and $\sqrt{3}$ are positive and $\sqrt{6} > \sqrt{3}$, this is the distance outside the sphere. 6. Final answer: $$\boxed{\sqrt{6} - \sqrt{3}}$$