Partial Fractions Associated with Quadratic Factors

Extract

The determination of the group of partial fractions corresponding to the powers of a linear factor x+b in the denominator of a rational function f (x)/{(x+b)ⁿφ(x)} is one of the comparatively rare algebraic problems whose theoretical solutions are feasible in practice. The numerator of (x+b)^n−r is the coefficient of y^r in the expansion of f(−b+y)/φ(−b+y), and the calculation of the group of numerators is effected by means of two Horner transformations followed by one division, which can of course be arranged synthetically; no operation need be taken further than the term in yⁿ⁻¹, whatever the degrees of f(x) and φ(x).