## Quartics

A quartic is a polynomialof degree 4. The polynomial may factorise with between one and four factors, some of which may be repeated. The different cases are

1. four distinct linear real factors :

2. a squared real linear factor and two different real linear factors :

3. two squared real linear factors:

4. a cubed factor and one real linear factor :

5. a real linear factor raised to the fourth power:

6. one real quadratic factor withand two real linear factors :

7. two real quadratic factors each with

If the coefficient ofis positive the graph tends to infinity as

Ifis fully factorised into real linear factors, then

For a distinct factorthe graph cuts the x - axis at | |

For each squared factorthe graph touches the x - axis at | |

Ifhas a cubed factorthe graph cuts the x - axis atand has a point of inflecion at | |

Ifhas a quadruple factor the graph touches the x - axis but is a minimum at that point. | |

If a quartic withhas one real quadratic factor withwe have | |

If a quartic withhas two real quadratic factors both with we have | |