Partial fraction expansion is a technique used in mathematics, specifically in calculus and algebra, to decompose a rational function into a sum of simpler fractions, called partial fractions. This technique is particularly useful when dealing with integrals of rational functions or solving linear differential equations.
The general form of a rational function is:
(
)
=
(
)
(
)
,
R(x)=
Q(x)
P(x)
,
where
(
)
P(x) and
(
)
Q(x) are polynomials, and the degree of
(
)
Q(x) is greater than the degree of
(
)
P(x).
Partial fraction expansion involves breaking down the rational function
(
)
R(x) into a sum of partial fractions in the form:
(
)
(
)
=
1
(
−
1
)
+
2
(
−
2
)
+
⋯
+
(
−
)
,
Q(x)
P(x)
=
(x−r
1
)
A
1
+
(x−r
2
)
A
2
+⋯+
(x−r
n
)
A
n
,
where
1
,
2
,
…
,
r
1
,r
2
,…,r
n
are distinct roots of the polynomial
(
)
Q(x), and
1
,
2
,
…
,
A
1
,A
2
,…,A
n
are constants that need to be determined.
The process of finding the constants
1
,
2
,
…
,
A
1
,A
2
,…,A
n
involves various methods, depending on the nature of the roots of
(
)
Q(x) and the degree of the polynomial. The main approaches include:
Distinct Linear Factors: If
(
)
Q(x) has distinct linear factors, the partial fractions can be written as above, and the constants can be found by equating coefficients.
Repeated Linear Factors: If
(
)
Q(x) has repeated linear factors, the partial fractions involve terms like
1
(
−
1
)
+
2
(
−
1
)
2
+
⋯
(x−r
1
)
A
1
+
(x−r
1
)
2
A
2
+⋯, and the constants are determined similarly.
Irreducible Quadratic Factors: If
(
)
Q(x) has irreducible quadratic factors, the partial fractions will include terms like
+
(
2
+
+
)
(ax
2
+bx+c)
Ax+B
, and the constants are found by comparing coefficients of
x and the constant term.
Once the partial fractions are found, the original rational function can be expressed as a sum of these simpler fractions, which often makes integration or solving differential equations easier.
Partial fraction expansion is widely used in various areas of mathematics and engineering, including calculus, differential equations, signal processing, and control systems. It simplifies complex expressions and allows for easier manipulation and analysis of functions involving rational terms.