52 Chapter 2Algebraic functions
EXAMPLE 2.26Dividex
31 − 17 x
21 + 116 x 1 − 110 byx 1 − 11.
The cubic in this example is (number) 1 larger than that in Example 2.25, and there is
no remainder of the division. It follows that the cubic can be factorized:
(see Example 2.23). In this case the fact that the rational function is not defined
at x 1 = 11 either before or after cancellation of the factor (x 1 − 11 ) has no practical
consequences and can be ignored.
EXAMPLE 2.27Dividex 1 + 12 byx 1 + 11.
In this case it is not necessary to resort to long division:
0 Exercises 57–60
2.7 Partial fractions
Consider
(2.30)
The quadratic denominator has been expressed as the product of two linear factors,
and the fraction has been decomposed into two simpler partial fractions. We will see
in Chapters 6 and 11 that the decomposition into partial fractions is an important tool
in the solution of some differential equations in the theory of reaction rates, and in
integration in general. A proper rational functionP(x) 2 Q(x), whose denominator can
be factorized, can always be decomposed into simpler partial fractions. The following
examples demonstrate three of the important simple cases. All others can be treated
in the same way.
EXAMPLE 2.28Two linear factors in the denominator
x
xx x x
−+
=
−
2
34
5
73
2
()()()() 74
1
32
1
12
1
1
1
2
2xx
xx x x
++
=
++
=
−
()( ) +
x
x
x
x
x
xx x
=
++
=
=+
2
1
11
1
1
1
1
1
1
1
1
()
xx x
x
xxx
x
xx
32 2271610
1
1610
1
61
−+−
−
=
−−+
−
=−+
()( )
00