Download Partial fraction decomposition algorithm

Partial fraction decomposition algorithm
Remarkable fact: Any proper rational function (i.e. degree of numerator <
degree of denominator) whose coefficients are real numbers can be written as
the sum of “simple” rational functions of the form
A
,
(ax + b)k
Bx + C
(ãx2
+ b̃x + c̃)k̃
,
where k, k̃ are non-negative integers and A, B, C, a, b, ã, b̃, c̃ are real numbers.
1. Let
f (x) =
p(x)
q(x)
be a rational function. If f is not proper (i.e. if degree p(x) ≥ degree q(x)),
use long division to obtain
N (x)
f (x) = f˜(x) +
,
D(x)
N (x)
where f˜(x) is a polynomial and
is a proper rational function.
D(x)
2. Factor D into a product of linear (degree 1) and irreducible quadratic
(degree 2) factors with real coefficients. Note: This is always possible in
theory by the Fundamental Theorem of Algebra.
3. For each factor of D of the form (ax + b)n , expect a decomposition of the
form
A1
A2
An
+
+ ··· +
.
ax + b (ax + b)2
(ax + b)n
4. For each factor of the form (ax2 + bx + c)m , expect a decomposition of the
form
B1 x + C1
B2 x + C2
Bm x + Cm
+
+ ··· +
.
ax2 + bx + c (ax2 + bx + c)2
(ax2 + bx + c)m
N (x)
equal to the sum of the terms gotten in steps 3 and 4.
D(x)
6. Multiply both sides of the equation in step 5 by D(x) and solve for the
unknown constants, either by substituting in “convenient” values of the
variable x or by equating the coefficients of terms with the same degree.
5. Set
1