Partial Fraction - Procedure

Procedure

Given two polynomials and, where the α_i are distinct constants and deg P < n, partial fractions are generally obtained by supposing that

and solving for the c_i constants, by substitution, by equating the coefficients of terms involving the powers of x, or otherwise. (This is a variant of the method of undetermined coefficients.)

A more direct computation, which is strongly related with Lagrange interpolation consists in writing

where is the derivative of the polynomial .

This approach does not account for several other cases, but can be modified accordingly:

• If deg P deg Q, then it is necessary to perform the Euclidean division of P by Q, using polynomial long division, giving P(x) = E(x) Q(x) + R(x) with deg R < n. Dividing by Q(x) this gives

and then seek partial fractions for the remainder fraction (which by definition satisfies deg R < deg Q).

• If Q(x) contains factors which are irreducible over the given field, then the numerator N(x) of each partial fraction with such a factor F(x) in the denominator must be sought as a polynomial with deg N < deg F, rather than as a constant. For example, take the following decomposition over R:

• Suppose Q(x) = (x − α)rS(x) and S(α) ≠ 0. Then Q(x) has a zero α of multiplicity r, and in the partial fraction decomposition, r of the partial fractions will involve the powers of (x − α). For illustration, take S(x) = 1 to get the following decomposition: