Partial fraction decomposition

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In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is a fraction such that the numerator and the denominator are both polynomials) is the operation that consists in expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.

The importance of the partial fraction decomposition lies in the fact that it provides an algorithm for computing the antiderivative of a rational function.

In symbols, one can use partial fraction expansion to change a rational fraction in the form

${\frac {f(x)}{g(x)}}$

where ƒ and g are polynomials, into an expression of the form

$\sum _{j}{\frac {f_{j}(x)}{g_{j}(x)}}$

where g_j (x) are polynomials that are factors of g(x), and are in general of lower degree. Thus, the partial fraction decomposition may be seen as the inverse procedure of the more elementary operation of addition of rational fractions, which produces a single rational fraction with a numerator and denominator usually of high degree. The full decomposition pushes the reduction as far as it will go: in other words, the factorization of g is used as much as possible. Thus, the outcome of a full partial fraction expansion expresses that fraction as a sum of fractions, where:

the denominator of each term is a power of an irreducible (not factorable) polynomial and
the numerator is a polynomial of smaller degree than that irreducible polynomial. To decrease the degree of the numerator directly, the Euclidean division can be used, but in fact if ƒ already has lower degree than g this isn't helpful.

Basic principles

The basic principles involved are quite simple; it is the algorithmic aspects that require attention in particular cases. On the other hand, the existence of a decomposition of a certain kind is an assumption in practical cases, and the principles should explain which assumptions are justified.

Assume a rational function R(x) = ƒ(x)/g(x) in one indeterminate x has a denominator that factors as

$g(x)=P(x)\cdot Q(x)\,$

over a field K (we can take this to be real numbers, or complex numbers). If P and Q have no common factor, then R may be written as

${\frac {A}{P}}+{\frac {B}{Q}}$

for some polynomials A(x) and B(x) over K. The existence of such a decomposition is a consequence of the fact that the polynomial ring over K is a principal ideal domain, so that

$CP+DQ=1\,$

for some polynomials C(x) and D(x) (see Bézout's identity).

Using this idea inductively we can write R(x) as a sum with denominators powers of irreducible polynomials. To take this further, if required, write:

${\frac {G(x)}{F(x)^{n}}}$

as a sum with denominators powers of F and numerators of degree less than F, plus a possible extra polynomial. This can be done by the Euclidean algorithm, polynomial case. The result is the following theorem:

Let ƒ and g be nonzero polynomials over a field K. Write g as a product of powers of distinct irreducible polynomials :
$g=\prod _{{i=1}}^{k}p_{i}^{{n_{i}}}.$

There are (unique) polynomials b and a_ij with deg a_ij < deg p_i such that

${\frac {f}{g}}=b+\sum _{{i=1}}^{k}\sum _{{j=1}}^{{n_{i}}}{\frac {a_{{ij}}}{p_{i}^{j}}}.$

If deg ƒ < deg g, then b = 0.

Therefore, when the field K is the complex numbers, we can assume that each p_i has degree 1 (by the fundamental theorem of algebra) the numerators will be constant. When K is the real numbers, some of the p_i might be quadratic, so in the partial fraction decomposition a quotient of a linear polynomial by a power of a quadratic might occur.

In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the p_i may be the factors of the square-free factorization of g. When K is the field of the rational numbers, as it is typically the case in computer algebra, this allows to replace factorization by greatest common divisor to compute the partial fraction decomposition.

Application to symbolic integration

For the purpose of symbolic integration, the preceding result may be refined into

Let ƒ and g be nonzero polynomials over a field K. Write g as a product of powers of pairwise coprime polynomials which have no multiple root in an algebraically closed field:
$g=\prod _{{i=1}}^{k}p_{i}^{{n_{i}}}.$

There are (unique) polynomials b and c_ij with deg c_ij < deg p_i such that

${\frac {f}{g}}=b+\sum _{{i=1}}^{k}\sum _{{j=2}}^{{n_{i}}}\left({\frac {c_{{ij}}}{p_{i}^{{j-1}}}}\right)'+\sum _{{i=1}}^{k}{\frac {c_{{i1}}}{p_{i}}}.$

where $X'$ denotes the derivative of $X.$

This reduces the computation of the antiderivative of a rational function to the integration of the last sum, with is called the logarithmic part, because its antiderivative is a linear combination of logarithms. In fact, we have

${\frac {c_{{i1}}}{p_{i}}}=\sum _{{\alpha _{j}:p_{i}(\alpha _{j})=0}}{\frac {c_{{i1}}(\alpha _{j})}{p'_{i}(\alpha _{j})}}{\frac {1}{x-\alpha _{j}}}.$

There are various methods to compute above decomposition. The one that is the simplest to describe is probably the so-called Hermite's method. As the degree of c_ij is bounded by the degree of p_i, and the degree of b is the difference of the degrees of f and g (if this difference is non negative; otherwise, b=0), one may write these unknowns polynomials as polynomials with unknown coefficients. Reducing the two members of above formula to the same denominator and writing that the coefficients of each power of x are the same in the two numerators, one gets a system of linear equations which can be solved to obtain the desired values for the unknowns coefficients.

Procedure

Given two polynomials $P(x)$ and $Q(x)=(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{n})$ , where the α_i are distinct constants and deg P < n, partial fractions are generally obtained by supposing that

${\frac {P(x)}{Q(x)}}={\frac {c_{1}}{x-\alpha _{1}}}+{\frac {c_{2}}{x-\alpha _{2}}}+\cdots +{\frac {c_{n}}{x-\alpha _{n}}}$

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

${\frac {P(x)}{Q(x)}}=\sum _{{i=1}}^{n}{\frac {P(\alpha _{i})}{Q'(\alpha _{i})}}{\frac {1}{(x-\alpha _{i})}}$

where $Q'$ is the derivative of the polynomial $Q$ .

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

If deg P $\geq$ 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

${\frac {P(x)}{Q(x)}}=E(x)+{\frac {R(x)}{Q(x)}},$

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:

${\frac {x^{2}+1}{(x+2)(x-1)\color {Blue}(x^{2}+x+1)}}={\frac {a}{x+2}}+{\frac {b}{x-1}}+{\frac {\color {OliveGreen}cx+d}{\color {Blue}x^{2}+x+1}}.$

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:

${\frac {P(x)}{Q(x)}}={\frac {P(x)}{(x-\alpha )^{r}}}={\frac {c_{1}}{x-\alpha }}+{\frac {c_{2}}{(x-\alpha )^{2}}}+\cdots +{\frac {c_{r}}{(x-\alpha )^{r}}}.$

Illustration

In an example application of this procedure, (3x + 5)/(1 − 2x)² can be decomposed in the form

${\frac {3x+5}{(1-2x)^{2}}}={\frac {A}{(1-2x)^{2}}}+{\frac {B}{(1-2x)}}.$

Clearing denominators shows that 3x + 5 = A + B(1 − 2x). Expanding and equating the coefficients of powers of x gives

5 = A + B and 3x = −2Bx

Solving for A and B yields A = 13/2 and B = −3/2. Hence,

${\frac {3x+5}{(1-2x)^{2}}}={\frac {13/2}{(1-2x)^{2}}}+{\frac {-3/2}{(1-2x)}}.$

Residue method

Over the reals

Partial fractions are used in real-variable integral calculus to find real-valued antiderivatives of rational functions. Partial fraction decomposition of real rational functions is also used to find their Inverse Laplace transforms. For applications of partial fraction decomposition over the reals, see

Application to symbolic integration, above
Partial fractions in Laplace transforms

General result

Let ƒ(x) be any rational function over the real numbers. In other words, suppose there exist real polynomials functions p(x) and q(x)≠ 0, such that

$f(x)={\frac {p(x)}{q(x)}}$

By dividing both the numerator and the denominator by the leading coefficient of q(x), we may assume without loss of generality that q(x) is monic. By the fundamental theorem of algebra, we can write

$q(x)=(x-a_{1})^{{j_{1}}}\cdots (x-a_{m})^{{j_{m}}}(x^{2}+b_{1}x+c_{1})^{{k_{1}}}\cdots (x^{2}+b_{n}x+c_{n})^{{k_{n}}}$

where a₁,..., a_m, b₁,..., b_n, c₁,..., c_n are real numbers with b_i² − 4c_i < 0, and j₁,..., j_m, k₁,..., k_n are positive integers. The terms (x − a_i) are the linear factors of q(x) which correspond to real roots of q(x), and the terms (x_i² + b_ix + c_i) are the irreducible quadratic factors of q(x) which correspond to pairs of complex conjugate roots of q(x).

Then the partial fraction decomposition of ƒ(x) is the following:

$f(x)={\frac {p(x)}{q(x)}}=P(x)+\sum _{{i=1}}^{m}\sum _{{r=1}}^{{j_{i}}}{\frac {A_{{ir}}}{(x-a_{i})^{r}}}+\sum _{{i=1}}^{n}\sum _{{r=1}}^{{k_{i}}}{\frac {B_{{ir}}x+C_{{ir}}}{(x^{2}+b_{i}x+c_{i})^{r}}}$

Here, P(x) is a (possibly zero) polynomial, and the A_ir, B_ir, and C_ir are real constants. There are a number of ways the constants can be found.

The most straightforward method is to multiply through by the common denominator q(x). We then obtain an equation of polynomials whose left-hand side is simply p(x) and whose right-hand side has coefficients which are linear expressions of the constants A_ir, B_ir, and C_ir. Since two polynomials are equal if and only if their corresponding coefficients are equal, we can equate the coefficients of like terms. In this way, a system of linear equations is obtained which always has a unique solution. This solution can be found using any of the standard methods of linear algebra. It can also be found with limits (see Example 5).

Examples

Example 1

$f(x)={\frac {1}{x^{2}+2x-3}}$

Here, the denominator splits into two distinct linear factors:

$q(x)=x^{2}+2x-3=(x+3)(x-1)$

so we have the partial fraction decomposition

$f(x)={\frac {1}{x^{2}+2x-3}}={\frac {A}{x+3}}+{\frac {B}{x-1}}$

Multiplying through by x² + 2x − 3, we have the polynomial identity

$1=A(x-1)+B(x+3)$

Substituting x = −3 into this equation gives A = −1/4, and substituting x = 1 gives B = 1/4, so that

$f(x)={\frac {1}{x^{2}+2x-3}}={\frac {1}{4}}\left({\frac {-1}{x+3}}+{\frac {1}{x-1}}\right)$

Example 2

$f(x)={\frac {x^{3}+16}{x^{3}-4x^{2}+8x}}$

After long-division, we have

$f(x)=1+{\frac {4x^{2}-8x+16}{x^{3}-4x^{2}+8x}}=1+{\frac {4x^{2}-8x+16}{x(x^{2}-4x+8)}}$

Since (−4)² − 4(8) = −16 < 0, x² − 4x + 8 is irreducible, and so

${\frac {4x^{2}-8x+16}{x(x^{2}-4x+8)}}={\frac {A}{x}}+{\frac {Bx+C}{x^{2}-4x+8}}$

Multiplying through by x³ − 4x² + 8x, we have the polynomial identity

$4x^{2}-8x+16=A(x^{2}-4x+8)+(Bx+C)x$

Taking x = 0, we see that 16 = 8A, so A = 2. Comparing the x² coefficients, we see that 4 = A + B = 2 + B, so B = 2. Comparing linear coefficients, we see that −8 = −4A + C = −8 + C, so C = 0. Altogether,

$f(x)=1+2\left({\frac {1}{x}}+{\frac {x}{x^{2}-4x+8}}\right)$

The following example illustrates almost all the "tricks" one would need to use short of consulting a computer algebra system.

Example 3

$f(x)={\frac {x^{9}-2x^{6}+2x^{5}-7x^{4}+13x^{3}-11x^{2}+12x-4}{x^{7}-3x^{6}+5x^{5}-7x^{4}+7x^{3}-5x^{2}+3x-1}}$

After long-division and factoring the denominator, we have

$f(x)=x^{2}+3x+4+{\frac {2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x}{(x-1)^{3}(x^{2}+1)^{2}}}$

The partial fraction decomposition takes the form

${\frac {2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x}{(x-1)^{3}(x^{2}+1)^{2}}}={\frac {A}{x-1}}+{\frac {B}{(x-1)^{2}}}+{\frac {C}{(x-1)^{3}}}+{\frac {Dx+E}{x^{2}+1}}+{\frac {Fx+G}{(x^{2}+1)^{2}}}$

Multiplying through by (x − 1)³(x² + 1)² we have the polynomial identity

${\begin{aligned}&{}\quad 2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x\\&=A(x-1)^{2}(x^{2}+1)^{2}+B(x-1)(x^{2}+1)^{2}+C(x^{2}+1)^{2}+(Dx+E)(x-1)^{3}(x^{2}+1)+(Fx+G)(x-1)^{3}\end{aligned}}$

Taking x = 1 gives 4 = 4C, so C = 1. Similarly, taking x = i gives 2 + 2i = (Fi + G)(2 + 2i), so Fi + G = 1, so F = 0 and G = 1 by equating real and imaginary parts. With C = G = 1 and F = 0, taking x = 0 we get A − B + 1 − E − 1 = 0, thus E = A − B.

We now have the identity

${\begin{aligned}&{}2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x\\&=A(x-1)^{2}(x^{2}+1)^{2}+B(x-1)(x^{2}+1)^{2}+(x^{2}+1)^{2}+(Dx+(A-B))(x-1)^{3}(x^{2}+1)+(x-1)^{3}\\&=A((x-1)^{2}(x^{2}+1)^{2}+(x-1)^{3}(x^{2}+1))+B((x-1)(x^{2}+1)-(x-1)^{3}(x^{2}+1))+(x^{2}+1)^{2}+Dx(x-1)^{3}(x^{2}+1)+(x-1)^{3}\end{aligned}}$

Expanding and sorting by exponents of x we get

${\begin{aligned}&{}2x^{6}-4x^{5}+5x^{4}-3x^{3}+x^{2}+3x\\&=(A+D)x^{6}+(-A-3D)x^{5}+(2B+4D+1)x^{4}+(-2B-4D+1)x^{3}+(-A+2B+3D-1)x^{2}+(A-2B-D+3)x\end{aligned}}$

We can now compare the coefficients and see that

${\begin{aligned}A+D&=&2\\-A-3D&=&-4\\2B+4D+1&=&5\\-2B-4D+1&=&-3\\-A+2B+3D-1&=&1\\A-2B-D+3&=&3,\end{aligned}}$

with A = 2 − D and −A −3 D =−4 we get A = D = 1 and so B = 0, furthermore is C = 1, E = A − B = 1, F = 0 and G = 1.

The partial fraction decomposition of ƒ(x) is thus

$f(x)=x^{2}+3x+4+{\frac {1}{(x-1)}}+{\frac {1}{(x-1)^{3}}}+{\frac {x+1}{x^{2}+1}}+{\frac {1}{(x^{2}+1)^{2}}}.$

Alternatively, instead of expanding, one can obtain other linear dependences on the coefficients computing some derivatives at x=1 and at x=i in the above polynomial identity. (To this end, recall that the derivative at x=a of (x−a)^mp(x) vanishes if m > 1 and it is just p(a) if m=1.) Thus, for instance the first derivative at x=1 gives

$2\cdot 6-4\cdot 5+5\cdot 4-3\cdot 3+2+3=A\cdot (0+0)+B\cdot (2+0)+8+D\cdot 0$

that is 8 = 2B + 8 so B=0.

Example 4 (residue method)

$f(z)={\frac {z^{{2}}-5}{(z^{2}-1)(z^{2}+1)}}={\frac {z^{{2}}-5}{(z+1)(z-1)(z+i)(z-i)}}$

Thus, f(z) can be decomposed into rational functions whose denominators are z+1, z−1, z+i, z−i. Since each term is of power one, −1, 1, −i and i are simple poles.

Hence, the residues associated with each pole, given by

${\frac {P(z_{i})}{Q'(z_{i})}}={\frac {z_{i}^{2}-5}{4z_{i}^{3}}}$ ,

are

$1,-1,{\tfrac {3i}{2}},-{\tfrac {3i}{2}}$ ,

respectively, and

$f(z)={\frac {1}{z+1}}-{\frac {1}{z-1}}+{\frac {3i}{2}}{\frac {1}{z+i}}-{\frac {3i}{2}}{\frac {1}{z-i}}$ .

Example 5 (limit method)

Limits can be used to find a partial fraction decomposition.^[1]

$f(x)={\frac {1}{x^{3}-1}}$

First, factor the denominator:

$f(x)={\frac {1}{(x-1)(x^{2}+x+1)}}$

The decomposition takes the form of

${\frac {1}{(x-1)(x^{2}+x+1)}}={\frac {A}{x-1}}+{\frac {Bx+C}{x^{2}+x+1}}$

As $x\to 1$ , the A term dominates, so the right-hand side approaches ${\frac {A}{x-1}}$ . Thus, we have

${\frac {1}{(x-1)(x^{2}+x+1)}}={\frac {A}{x-1}}$

$A=\lim _{{x\to 1}}{{\frac {1}{x^{2}+x+1}}}={\frac {1}{3}}$

As $x\to \infty$ , the right-hand side is

$\lim _{{x\to \infty }}{{\frac {A}{x-1}}+{\frac {Bx+C}{x^{2}+x+1}}}={\frac {A}{x}}+{\frac {Bx}{x^{2}}}={\frac {A+B}{x}}.$

${\frac {A+B}{x}}=\lim _{{x\to \infty }}{{\frac {1}{x^{3}-1}}}=0$

Thus, $B=-{\frac {1}{3}}$ .

At $x=0$ , $-1=-A+C$ . Therefore, $C=-{\frac {2}{3}}$ .

The decomposition is thus ${\frac {{\frac {1}{3}}}{x-1}}+{\frac {-{\frac {1}{3}}x-{\frac {2}{3}}}{x^{2}+x+1}}$ .

The role of the Taylor polynomial

The partial fraction decomposition of a rational function can be related to Taylor's theorem as follows. Let

$P(x),Q(x),A_{1}(x),\dots ,A_{r}(x)$

be real or complex polynomials; assume that

$\textstyle Q=\prod _{{j=1}}^{{r}}(x-\lambda _{j})^{{\nu _{j}}},$

that

$\textstyle \deg(P)<\deg(Q)=\sum _{{j=1}}^{{r}}\nu _{j},$

and that

$\textstyle \deg A_{j}<\nu _{j}{\text{ for }}j=1,\dots ,r.$

Define also

$\textstyle Q_{i}=\prod _{{j\neq i}}(x-\lambda _{j})^{{\nu _{j}}}={\frac {Q}{(x-\lambda _{i})^{{\nu _{i}}}}}{\text{ for }}i=1,\dots ,r.$

Then we have

${\frac {P}{Q}}=\sum _{{j=1}}^{{r}}{\frac {A_{j}}{(x-\lambda _{j})^{{\nu _{j}}}}}$

if, and only if, for each $\textstyle i$ the polynomial $\textstyle A_{i}(x)$ is the Taylor polynomial of $\textstyle {\frac {P}{Q_{i}}}$ of order $\textstyle \nu _{i}-1$ at the point $\textstyle \lambda _{i}$ :

$A_{i}(x):=\sum _{{k=0}}^{{\nu _{i}-1}}{\frac {1}{k!}}\left({\frac {P}{Q_{i}}}\right)^{{(k)}}(\lambda _{i})\ (x-\lambda _{i})^{k}.$

Taylor's theorem (in the real or complex case) then provides a proof of the existence and uniqueness of the partial fraction decomposition, and a characterization of the coefficients.

Sketch of the proof: The above partial fraction decomposition implies, for each 1 ≤ i ≤ r, a polynomial expansion

${\frac {P}{Q_{i}}}=A_{i}+O((x-\lambda _{i})^{{\nu _{i}}})\qquad$ , as $x\to \lambda _{i};$

so $\textstyle A_{i}$ is the Taylor polynomial of $\textstyle {\frac {P}{Q_{i}}}$ , because of the unicity of the polynomial expansion of order $\textstyle \nu _{i}-1$ , and by assumption $\textstyle \deg A_{i}<\nu _{i}$ .

Conversely, if the $\textstyle A_{i}$ are the Taylor polynomials, the above expansions at each $\textstyle \lambda _{i}$ hold, therefore we also have

$P-Q_{i}A_{i}=O((x-\lambda _{i})^{{\nu _{i}}})\qquad$ , as $x\to \lambda _{i},$

which implies that the polynomial $\textstyle P-Q_{i}A_{i}$ is divisible by $\textstyle (x-\lambda _{i})^{{\nu _{i}}}.$

For $\textstyle j\neq i$ also $\textstyle Q_{j}A_{j}$ is divisible by $\textstyle (x-\lambda _{i})^{{\nu _{i}}}$ , so we have in turn that $\textstyle P-\sum _{{j=1}}^{{r}}Q_{j}A_{j}$ is divisible by $\textstyle Q$ . Since $\textstyle \deg \left(P-\sum _{{j=1}}^{{r}}Q_{j}A_{j}\right)<\deg(Q)$ we then have $\textstyle P-\sum _{{j=1}}^{{r}}Q_{j}A_{j}=0$ , and we find the partial fraction decomposition dividing by $\textstyle Q$ .

Fractions of integers

The idea of partial fractions can be generalized to other integral domains, say the ring of integers where prime numbers take the role of irreducible denominators. For example:

${\frac {1}{18}}={\frac {1}{2}}-{\frac {1}{3}}-{\frac {1}{3^{2}}}.$

Notes

↑ Bluman, George W. (1984). Problem Book for First Year Calculus. New York: Springer-Verlag. pp. 250–251.

References

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External links

Weisstein, Eric W., "Partial Fraction Decomposition", MathWorld.
Blake, Sam. "Step-by-Step Partial Fractions".
Make partial fraction decompositions with Scilab.

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