two numerators, one gets a system of linear equations which can be solved to obtain the desired (unique) values for the unknown coefficients. Procedure Given two polynomials and , where the αi are distinct constants and , explicit expressions for partial fractions can be obtained by supposing that and solving for the ci 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. After both sides of the equation are multiplied by Q(x), one side of the equation is a specific polynomial, and the other side is a polynomial with undetermined coefficients. The equality is possible only when the coefficients of like powers of x are equal. This yields n equations in n unknowns, the ck.) A more direct computation, which is strongly related to Lagrange interpolation, consists of writing where is the derivative of the polynomial . The coefficients of are called the residues of f/g. This approach does not account for several other cases, but can be modified accordingly: If then it is necessary to perform the Euclidean division of P by Q, using polynomial long division, giving with . Dividing by Q(x) this gives and then seek partial fractions for the remainder fraction (which by definition satisfies ). 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 , rather than as a constant. For example, take the following decomposition over R: Suppose and , that is is a root of of multiplicity . In the partial fraction decomposition, the first powers of will occur as denominators of the partial fractions (possibly with a zero numerator). For example, if the partial fraction decomposition has the form Illustration In an example application of this procedure, can be decomposed in the form Clearing denominators shows that . Expanding and equating the coefficients of powers of gives Solving this system of linear equations for and yields . Hence, Residue method Over the complex numbers, suppose f(x) is a rational proper fraction, and can be decomposed into Let then according to the uniqueness of Laurent series, aij is the coefficient of the term in the Laurent expansion of gij(x) about the point xi, i.e., its residue This is given directly by