Simplifying algebraic fractions (GCSE algebra)

Towards the end of a GCSE paper, you’re quite frequently asked to simplify an algebraic fraction like:

$\frac{4 x^{2} + 12 x - 7}{2 x^{2} + 5 x - 3}$

Hold back the tears, dear students, hold back the tears. These are easier than they look. There’s one thing you need to know: algebraic fractions are happiest when they’re in brackets.

If you’re a regular reader, you’ll know how to put quadratic expressions in brackets - check out this article, or this one if you prefer.

To factorise the top, you convert it to $X^{2} + 12 X - 28$ (shuttling the 4) and factorise: $(X + 14) (X - 2)$ . You then move a two from the 14 to the opposite $X$ , and a two from the -2 to to opposite X to get $(2 x - 7) (2 x - 1)$ . Lovely.

The bottom works much the same way: it becomes $X^{2} + 5 X - 6$ , or $(X + 6) (X - 1)$ . Shuttle a 2 from the -6 to the other X to get $(x - 3) (2 x - 1)$ .

The fraction is now $\frac{(2 x - 7) (2 x - 1)}{(x - 3) (2 x - 1)}$ . Aha! There’s a common factor of $(2 x - 1)$ on the top and the bottom, so we can cancel that to get: $\frac{2 x - 7}{x - 3}$ - which is our fully-simplified answer.

Once you’ve done a handful of these, you’ll start to get a Pavlovian response to this kind of algebraic fraction, and dive straight in!

One thing to look out for is difference of two squares, which comes up once in a while and catches some students out. But you’re smarter than that, right? Right.

* Edited 2014-09-11 for clarity and to fix LaTeX errors.

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