This is called implicitization . Unfortunately, the answer is negative. This is the result of uniformization theorems . Due to this fact, implicit polynomials can describe more curves than polynomial and rational parametric forms. The reason is simple.
Parametric curves are easy to use.
While implicit polynomial curves are powerful and can describe more curves, many of its important properties are difficult to compute and implicit curves are in general difficult to use. We shall cover implicit curves and surfaces later.