Understanding Engineering Mathematics

Just as we can simplify the fraction^24 to its ‘lowest’ form^12 , so we may be able to ‘cancel
down’ algebraic fractions – but with slightly more care. For example on cancelling thex
on top and bottom you may be happy with

x(x+ 1 )

x

=x+ 1

but remember that the functions on each side arenotthesameforallvaluesofx.Whereas
the right-hand side exists for all values ofx, the left-hand side is not defined forx=0.
Provided you are careful about such things, algebraic cancellations present little difficulty.

Examples
x^2 −x
x− 1

=

x(x− 1 )

x− 1

=x(x    = 1 )

3 x− 6

x^2 −x− 2

=

3 (x− 2 )

(x+ 1 )(x− 2 )

=

3

x+ 1

(x  =− 1 , 2 )

In algebraic fractions it is perhaps not so much what youcan dothat needs emphasising,

but what youcan’t do. For example, presented with something like

x+ 1

x+ 2

, you cannot

‘cancel’ thex’s as follows:

‘x+^1

x+ 2

=

x+ 1

x+ 2

=

1 + 1

1 + 2

=

2

3

’

The fact is that

x+ 1

x+ 2

cannot be ‘simplified’ further – leave it as it is. Tread warily with

algebraic fractions – check each step, with numerical examples if need be.
We now consider addition and subtraction of rational functions. First, get used to adding
fractions with the same denominator:

a

c

+

b

c

=

a+b

c

This is the only way you can add fractions directly – when the denominators are the
same (13

➤

).

Example
3
x− 1

−

7

x− 1

=

3 − 7

x− 1

=

− 4

x− 1

To add fractions with different denominators we have to make each denominator the same
by multiplying top and bottom by an appropriate factor. We use the result for constructing
equivalent fractions:

1

a

=

1

a

×

b

b

=

b

ab

(a,b    = 0 ) e.g.

1

3

=

5

15