Finding the Equation of a Quartic
The equation of a quartic can have up to five constants:
This means we need five conditions to determine the equation of the quartic. In fact we always need five condtions, but some of these may be hidden. If the equation of the quartic includes a repeated factor
then the corresponding condition would be ' the graph of the quartic is a tangent at
'. This statement includes two conditions.
1. is a tangent
2. at
Taking this sort of thing into account, we can find the equation of the quartic.
Example: Find the quartic which touches the x - axis at 3, cuts it at -2 and also passes
through
and
1. ' touches the x - axis at 3' implies the quartic is a tangent at
so includes a factor
2. ' cuts it at -2' implies a factor
The quartic must then take the form
passes throughso
(1)
passes through
so
(2)
(1) divided by (2) gives
Then from (2)
The equation of the quartic is