Understanding Engineering Mathematics

2.2.8 Algebra of rational functions

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Rational functions often cause a lot of problems for beginners in mathematics. There are
some common errors that occur frequently, such as

‘ 1

a

+

1

b

=

1

a+b

’

To clear these up we will approach the algebra of rational functions gradually and system-
atically. First, to avoid such errors as that above, it may be helpful for you to get into the
habit of checking such results numerically. For example, puttinga=b=1 in the above
gives the nonsense

‘ 1

1

+

1

1

= 1 + 1 = 2 =

1

1 + 1

=

1

2

’

Algebraic fractions behave much like numerical fractions. For example, multiplication is
straightforward:

a

b

×

c

d

=

ac

bd

wherea,b,c,dmay be any polynomials.
Dividing by a rational function requires a little care, but essentially depends on the
fact that

1

1 /a

=a(a    = 0 )

• the reciprocal of the reciprocal undoes the reciprocal – as can be seen by multiplying
  top and bottom on the left-hand side bya. In general, division is carried out by inverting

the dividing fraction and multiplying, thus:

a

b

/

c

d

=

a

b

×

d

c

=

ad

bc

. Again, this can be

seen by multiplying top and bottom of the left-hand expression by

d

c

.

Example

Multiply and divide the functionsx+1,

1

x

, stating when the corresponding operations are

permissible.

We can have the product(x+ 1 )×

1

x

=

x+ 1

x

= 1 +

1

x

forx    =0.

For division we can have:

x+ 1

1 /x

=(x+ 1 )×

1

1 /x

=(x+ 1 )×x

=x(x+ 1 ) forx  = 0

or

1 /x

x+ 1

=

1

x

×

1

x+ 1

=

1

x(x+ 1 )

forx    =0or− 1