Cubic curve in projective space

Question

Is it true that every cubic curve in $\mathbb{P}^3$, which is not contained in a plane, can be parametrized by polynomials? $\\\\\\\\$

Answer 1

See the paper On Varieties of Minimal Degree (A Centennial Account) by Eisenbud–Harris.

The first theorem in that paper says:

Theorem 1: If $X \subset \mathbf P^r$ is a variety of minimal degree, then $X$ is a cone over a smooth such variety. If $X$ is smooth and $\text{codim } X>1$, then $X \subset \mathbf P^r$ is either a rational normal scroll or the Veronese surface $\mathbf P^2 \subset \mathbf P^5$.

Now a cubic curve in $\mathbf P^3$ is of minimal degree. So by Theorem 1, if it's a cone, then it's a union of three lines, which is certainly rational (i.e. parametrised by polynomials); if not, then it is smooth, hence a rational normal curve, which again is what you want.