The article considers numerical algorithms for determining the coefficients of polynomials with a fixed leading coefficient, the algorithms supplying a minimum deviation from zero in a minimax norm with a given weight function. The polynomials serve as a useful tool in many numerical methods, in particular, in the Lanczos’ tau method which provides an approximate numerical analytic solution of ordinary differential equations with coefficients as polynomials in the independent variable. The well-known Chebyshev polynomials determined analytically are the special case of such polynomials, however, in most cases of weight functions, such polynomials can only be determined and tabulated numerically.