Algebraic Fractions
Algebraic fractions have properties which are the same as
those for numerical fractions, the only difference being that the
the numerator (top) and denominator (bottom) are both algebraic
expressions.
Example 1
Simplify each of the following fractions.
Solution
N.B. The cancellation in (b) is allowed since
x is a common factor of the numerator and the denominator .
Sometimes a little more work is necessary before an algebraic
fraction can be reduced to a simpler form.
Example 2
Simplify the algebraic fraction
Solution
In this case the numerator and denominator can be factored
into two terms, thus
With this established the simplification proceeds as follows:
(cancelling (x1))
Exercise 1
Simplify each of the following algebraic fractions.
(a)
(b)
Solution
(a) The fraction is .
This time, instead of expanding the factors, it is
easier to use the rule for powers
.
Thus
(b) In this case, some initial factorisation
is needed.
Thus
where the factor ( y + 5) has been cancelled.
Quiz
Which of the following is a simplified version of
Solution
The numerator and denominator respectively factorise as
so that
where the factor ( t  1) has been cancelled from the first
equation.
So far, simplification has been achieved by cancelling common
factors from the numerator and denominator. There are fractions
which can be simplified by multiplying the numerator and
denominator by an appropriate common factor, thus obtaining an
equivalent, simpler expression.
Example 3
Simplify the following fractions.
Solution
(a) In this case, multiplying both the
numerator and the denominator by 4 gives:
(b) To simplify this expression, multiply the
numerator and denominator by {\f2 x. Thus
Exercise 2
Simplify each of the following algebraic fractions.
(a)
(b)
Solution
(a) The fraction is simplified by multiplying
both the numerator and the denominator by 2 .
(b) In this case, since the numerator
contains the fraction 1/3 and the denominator contains the
fraction 1/2 , the common factor needed is 2 Ã— 3 = 6 . Thus
Quiz
Which of the following is a simplified version of
Solution
For , the common multiplier is ( x + 1) . Multiplying the
numerator and the denominator by this gives:
