An extension L/K of skew fields is called a left polynomial extension with polynomial generator q if it has a left basis of the form 1, q, q^2,...q^(n-1) for some n. This notion of left polynomial extension is a generalisation of the notion of pseudo-linear extension, known from the literature. In this paper, we show that any polynomial which is the minimal polynomial over K of some element in an ... [Show full abstract] extension of K occurs as the polynomial related to a polynomial generator of some polynomial extension. We also prove that every left cubic extension is a left polynomial extension. Furthermore, we give a characterisation of all left cubic extensions which have right degree 2 and construct an example of such a left cubic extension which is not pseudo-linear and which cannot be obtained as a homomorphic image of some form of a skew polynomial ring. Moreover, we give a classification of all cubic Galois extensions and construct examples of them. It is proved that any quartic central extension of a noncommutative ground field is a polynomial extension. A nontrivial example of a quartic central polynomial extension with a noncommutative centralizer is also described. A characterisation is given of a right predual extension of a right polynomial extension in terms of the existence of certain separate zeros. As a corollary, a characterisation is derived for polynomial extensions which are Galois extensions in terms of the existence of separate zeros. Finally, it is proved that any right polynomial extension has a dual extension that is left polynomial.