54 Chapter 2Algebraic functions
EXAMPLE 2.30A repeated linear factor in the denominator
and it follows thatA 1 = 13 andB 1 = 1 − 8.
0 Exercise 65
In the general case, the decomposition of a proper rational functionP(x) 2 Q(x)in
partial fractions depends on the nature of the roots of the denominatorQ(x)(see
Section 2.5 on the roots of the general polynomial).
(i) All the roots are real.
In this caseQ(x)can be factorized as the product of real linear factors; if Qhas degree
nthen
Q(x) 1 = 1 a(x 1 − 1 x
1)(x 1 − 1 x
2)1-1(x 1 − 1 x
n) (2.31)
wherex
1,x
2, =,x
nare the roots. If all the roots are different thenP(x) 2 Q(x)can be
decomposed into the sum of nsimple fractions, as in Examples 2.28 and 2.29:
(2.32)
If some of the roots are equal then there are additional terms, as in Example 2.30,
with powers of the linear factor in the denominator. For example, ifx
11 = 1 x
21 = 1 x
3in
Q(x)then
(2.33)
(ii) Some of the roots are complex.
If complex numbers are allowed then the discussion of (i) above is applicable. If
complex numbers are disallowedthen the denominatorQ(x)can be factorized as the
product of linear factors, one for each real root, and one or more real quadratic factors,
one for each pair of complex conjugate roots (Section 8.2). The decomposition of the
rational function in partial fractions then contains, in addition to the terms discussed
in (i) above, one fraction for each quadratic factor, of the form
(2.34)
ax b
xpxq
++
2=
++
−
−
++
−
ax bx c
xx
c
xx
c
xx
nn21344()
Px
Qx
c
xx
c
xx
c
xx
c
xx
()
()
()()
=
−
−
−
−
1121231344+
−
c
xx
nnPx
Qx
c
xx
c
xx
c
xx
nn()
()
=
−
−
++
−
112231
3
3
33
3
222x
x
A
x
B
x
Ax B
x
Ax
=
=
++
=
() ()
()
()
( 33
3
2AB
x
)
()