# Using Rational Expressions

A Rational Expression is the ratio of two polynomials:

You can read more on our page on Rational Expressions

## Using Rational Expressions

Using Rational Expressions is very similar to Using Rational Numbers (you might like to read that first).

## Adding Rational Expressions

The easiest way to add Rational Expressions is to use the common denominator method:

Like in this example:

### Example:

2 | + | 3 | = | 2·(x+1) + (x-2)·3 |

x-2 | x+1 | (x-2)(x+1) |

Which can then be simplified to:

= | 2x+2 + 3x-6 | = | 5x-4 |

x^{2}+x-2x-2 |
x^{2}-x-2 |

## Subtracting Rational Expressions

Subtracting is just like Adding:

### Example:

2 | - | 3 | = | 2·(x+1) - (x-2)·3 |

x-2 | x+1 | (x-2)(x+1) |

Which can then be simplified to:

= | 2x+2 - (3x-6) | = | -x+8 |

x^{2}+x-2x-2 |
x^{2}-x-2 |

## Multiplication

To multiply two Rational Expressions, just multiply the tops and bottoms separately, like this:

### Example:

2 | × | 3 | = | 2·3 |

x-2 | x+1 | (x-2)(x+1) |

Which can then be simplified to:

= | 6 |

x^{2}-x-2 |

## Division

To divide two Rational Expressions, first flip the second expression over (make it a reciprocal) and then do a multiply like above:

### Example:

First flip the second one over and make it a multiply:

2 | / | 3 | = | 2 | × | x+1 |

x-2 | x+1 | x-2 | 3 |

Then do the multiply:

2 | × | x+1 | = | 2(x+1) |

x-2 | 3 | 3(x-2) |

## Simplifying

When simplifying a rational function be careful to respect where the lower polynomial is equal to zero

### Example:

is undefined where x=-1

Its Domain (the values that can go into the expression) does not include -1

Now, we can factor x^{2}-1 into (x-1)(x+1) so we get:

It is now tempting to cancel (x+1) from top and bottom to produce:

x - 1

Its Domain now **does** include -1

But it is now a different function because it has a different Domain.