The author studies biorthogonal polynomials as defined in the book by \textit{C. Brezinski} [``Biorthogonality and its applications to numerical analysis'' (1991; Zbl 0757.41001)]. Let \(\{L_i\}\) denote a sequence of linear functionals on the space of polynomials over \({\mathbb C}\) and introduce the determinants \[ N^{(i,j)}_{n+1}(x)=\left|\begin{matrix} L_i(x^j) & \cdots & L_i(x^{j+n}) \\ \vdots & & \vdots \\ L_{i+n-1}(x^j) & \cdots & L_{i+n-1}(x^{j+n}) \\ 1 & \cdots & x^n\end{matrix}\right|, \] and \[ D^{(i,j)}_n=\left|\begin{matrix} L_i(x^j) & \cdots & L_i(x^{j+n-1}) \\ \vdots & & \vdots \\ L_{i+n-1}(x^j) & \cdots & L_{i+n-1}(x^{j+n-1})\end{matrix}\right|. \] The polynomials are given by \[ P^{i,j}_n(x):={N^{(i,j)}_{n+1}(x) \over D^{(i,j)}_n} \] and satisfy the biorthogonality relations \[ L_p(x^jP_n^{i,j}(x))=0,\;p=i,\ldots, i+n-1. \] Particular cases are the so-called \textit{vector orthogonal polynomials} (connected with a generalization of the Padé type approximant). As the explicit calculation using the determinantal expression is not feasible for increasing \(n\), the methods of using a fixed algorithm and using simultaneous algorithms are considered. Furthermore, relations between three biorthogonal polynomials are studied and all relations of a certain type are determined, leading to 12 relations (not all linearly independent). Finally, the coefficients in any three independent relations are looked into. They satisfy identities that can be used to derive a generalization of the famous \textit{QD-algorithm} due to Rutishauser.