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# Working with rational functions or expressions

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## What are "rational functions?"

A rational function is one that is expressed as a fraction - this is usually what "rational" means in mathematics.

They are also referred to as **rational expressions.**

You can expect to see any types of functions involved: linear, quadratic, cubic... The key to any function being a **rational function** is the fact that one expression is divided by the other.

## Simplifying rational functions

To simplify a rational function, the best general advice I can give is: *Factorise everything and cancel down*. This should always be your first step!

**Tip:** Don't forget to keep an eye out for the difference of two squares!

## Useful links:

- Notes: Rational expressions

Further notes on working with rational expressions. - Practice: Difference of two squares

More on the difference of two squares, with practice questions.

## Example:

Looking at the first example to the right, if we factorise **2x ^{2} + 8x + 6** we get

**(2x + 2)(x +3)**, doing the same to

**x**gives

^{2}+ 7x + 12**(x + 3)(x + 4)**. If we divide these two rewritten expressions, we can see that common factors cancel out - we can remove (x + 3) from the numerator and from the denominator.

This would leave (2x + 2) over (x + 4) as our simplifed rational function.

## Example using difference of two squares:

The second example looks harder, until you realise that (x^{2} - 9) can be factorised using the *difference of two squares* method; as it's (x^{2} - 3^{2}) we can rewrite it as (x + 3)(x - 3).

After that, simply cancel down as before to leave (x + 4) over (x - 3).

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