# Using Rational Expressions

A Rational Expression is the ratio of two polynomials:

## Using Rational Expressions

Using Rational Expressions is very similar to Using Rational Numbers (you may 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 |

x^{2}+x−2x−2 |

= | 5x−4 |

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^{2}+x−2x−2 |

= | −x+8 |

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.