Simplifying Algebraic Fractions (Edexcel IGCSE Maths)

Spec point — 1.2A

Simplifying algebraic fractions

What is an algebraic fraction?

• An algebraic fraction is a fraction with an algebraic expression on the top (numerator) and/or the bottom (denominator)

How do you simplify an algebraic fraction?

• If possible, factorise fully the top and bottom

  • E.g. $\frac{2 x}{x^{2} + 3 x} = \frac{2 x}{x ( x + 3 )}$

• Cancel common factors

  • This factor may be a single term

    • E.g. $\frac{ \cancel{x} ( 5 x - 1 )}{4 \cancel{x} } = \frac{5 x - 1}{4}$

  • It could also be a common bracket

    • E.g. $\frac{x \cancel{( x + 2 )} }{ \cancel{( x + 2 )} ( x - 1 )} = \frac{x}{x - 1}$

• A common mistake is to cancel a factor that is not common to all terms in either the top or the bottom of a fraction

  • E.g. The fraction $\frac{6x}{x+1}$ cannot be simplified

    • $x$ is not common to all terms in the bottom of the fraction
    • and the expression on the bottom cannot be factorised

Exam Hint When asked to simplify an algebraic fraction, factorise top and bottom.

It is very likely that one of the factors will be the same on the top and the bottom.

Use this fact to help you if one of the expressions is difficult to factorise!

Worked Example Simplify $\frac{4x+6}{2x^{2}-7x-15}$

Answer:

Factorise the top, by using 2 as a common factor

$\frac{2 ( 2 x + 3 )}{2 x^{2} - 7 x - 15}$

Factorise the bottom using your preferred method Using the fact that the top factorised to $( 2 x + 3 )$ may help!

$\frac{2 ( 2 x + 3 )}{( 2 x + 3 ) ( x - 5 )}$

The common factors on the top and bottom reduce to 1 (cancel out)

$\frac{2 \cancel{( 2 x + 3 )} }{ \cancel{( 2 x + 3 )} ( x - 5 )}$

$\frac{2}{(x-5)}$