  The main goal of the paper is to introduce methods which compute B\'ezier
curves faster than Casteljau's method does. These methods are based on the
spectral factorization of a $n\times n$ Bernstein matrix, $B^e_n(s)=
P_nG_n(s)P_n^{-1}$, where $P_n$ is the $n\times n$ lower triangular Pascal
matrix. So we first calculate the exact optimum positive value $t$ in order to
transform $P_n$ in a scaled Toeplitz matrix, which is a problem that was
partially solved by X. Wang and J. Zhou (2006). Then fast Pascal matrix-vector
multiplications and strategies of polynomial evaluation are put together to
compute B\'ezier curves. Nevertheless, when $n$ increases, more precise Pascal
matrix-vector multiplications allied to affine transformations of the vectors
of coordinates of the control points of the curve are then necessary to
stabilize all the computation.
