## Simplifying Algebraic Fractions

An algraic fraction is an fraction in which the numerator or denominator or both, include x terms.

are all algebraic fractions.

Apart from the last term, these fractions cannot be simplified. For an algebraic fraction to be simplified, the numerator or denominator or both must factorise so that there are common factors in numerator and denominator. These will then cancel, leaving a simpler expression.

The last example above factorises asThere is a common factor in numerator and denominator ofWe cancel this to obtain the simpler expression

The following examples all show how numerator and denominator can be factorised and factors cancelled to leave a simpler expression.

It is important to realise that not all expressions simplify. The second example above only sim[lifies because all the terms in numerator and denominator have a common factor of 2, which then cancels throughout.

does not simplify because there is no factor common to each term in numerator and denominator..

does not simplify because even though numerator and denominator both factorise, they do not do so with a common factor in numerator and denominator.