is an irreducible polynomial. If , one can decompose further, by using that an irreducible polynomial is a square-free polynomial, that is, is a greatest common divisor of the polynomial and its derivative. If is the derivative of , Bézout's identity provides polynomials and such that and thus Euclidean division of by gives polynomials and such that and Setting one gets with Iterating this process with in place of leads eventually to the following theorem. Statement The uniqueness can be proved as follows. Let . All together, and the have coefficients. The shape of the decomposition defines a linear map from coefficient vectors to polynomials of degree less than . The existence proof means that this map is surjective. As the two vector spaces have the same dimension, the map is also injective, which means uniqueness of the decomposition. By the way, this proof induces an algorithm for computing the decomposition through linear algebra. If is field of complex numbers, the fundamental theorem of algebra implies that all have degree one, and all numerators are constants. When is the field of real numbers, some of the may be quadratic, so, in the partial fraction decomposition, quotients of linear polynomials by powers of quadratic polynomials may also occur. In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the may be the factors of the square-free factorization of . When is the field of rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor computation for computing a partial fraction decomposition. Application to symbolic integration For the purpose of symbolic integration, the preceding result may be refined into This reduces the computation of the antiderivative of a rational function to the integration of the last sum, which is called the logarithmic part, because its antiderivative is a linear combination of logarithms. There are various methods to compute decomposition in the Theorem. One simple way is called Hermite's method. First, b is immediately computed by Euclidean division of f by g, reducing to the case where deg(f) < deg(g). Next, one knows deg(cij) < deg(pi), so one may write each cij as a polynomial with unknown coefficients. Reducing the sum of fractions in the Theorem to a common denominator, and equating the coefficients of each power of x in the