第八部分
Partial Fractions (部分分式)
If f ( x) and g ( x ) are polynomials and the degree of f ( x) is less than the degree of
g ( x ) (註:表示有理函數 f ( x) g ( x) 為真分式), then it is possible to write f ( x) g ( x) as a
sum of rational expressions whose denominators involve powers of polynomials of degree
not greater than two. i.e.,
f ( x)
 F1 ( x)  F2 ( x) 
g ( x)
 Fk ( x)
where each Fi ( x) has one of the forms
A
 px  q 
m
or
 ax
Cx  D
2
 bx  c 
n
for some
nonnegative integers m and n , and where ax 2  bx  c is irreducible ( 註 : 表 示
a x2  b 
x 不可因式分解).
c
The expression on the right side of the equation is called the
partial fraction decomposition (分解) of f ( x) g ( x) .
(1) Write the partial fraction decomposition of f ( x) 
1
and find f ( n ) ( x) .
x  5x  6
2
(sol): x 2  5x  6  ( x  2)( x  3)
1
x  5x  6
1
A
B



( x  2)( x  3) x  2 x  3
 1  A( x  3)  B( x  2)
f (x )
2
① x  2 代入  1   A  A  1
② x  3 代入  1  B  B  1
 The partial fraction decomposition of f ( x) 
1
1
1


x  5x  6 x  2 x  3
2
and
f ( n ) ( x)    1  n !  x  2 
n

  1  n !   x  2 
n
  1  n !  x  3
( n 1 )
 ( n 1)
n
  x  3
 ( n 1)
1

 n (
1 )
(2) Write the partial fraction decomposition of f ( x) 
4 x 2  13x  9
and find f ( n ) ( x) .
x3  2 x 2  3x
(sol): x3  2 x 2  3x  x  x 2  2 x  3  x( x  1)( x  3)
4 x 2  13 x  9
x3  2 x 2  3x
4 x 2  13 x  9 A
B
C

 

x( x  1)( x  3) x x  1 x  3
f ( x) 
 4 x 2  13 x  9  A( x  1)( x  3)  Bx( x  3)  Cx( x  1)
① x  0 代入  9  3A  A  3
② x  1 代入  8  4B  B  2
③ x  3 代入  12  12C  C  1
 The partial fraction decomposition of f ( x) 
4 x 2  13x  9 3
2
1
 

3
2
x  2 x  3x x x  1 x  3
and
1 )
f ( n )( x ) 3   1  n  x! n( 
 2
n


n
1n   x !
)
  1  n! 3  x  ( n  1 
2   x  1
n

1  
(n 1 )
  x  3
( n 1 )
 n (


n
1 )
＃
2
 n1  x  ! 
 n (
1 )
3