One may think of the coefficients ck as representing a power series expansion of any general function. In the rational function, one has to set a scale, usually by defining b0 = 0. This leaves m + n + 1 unknowns, the coefficients ai and bi, for which it is unproblematic to solve: the expression is multiplied with the denominator of the rational function, giving on both sides of the equation polynomials containing the unknown coefficients; one equates all terms with the same power of x to obtain the solution.
Padé approximations are useful for representing unknown functions with possible poles, i.e. with denominators tending towards zero. For a discussion and algorithm, see [Press95], also [Wong92].