For a given monic quartic equation
 |
(1)
|
the resolvent cubic is the monic cubic polynomial
 |
(2)
|
where the coefficients 
are given in terms of the 
by
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
The roots 
, 
, and 
of 
are given in terms of the roots 
, 
, 
, and 
of 
by
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
The resolvent cubic of a quartic equation can be used to solve for the roots of the quartic in terms of the roots of the cubic, which can in turn be solved for using the cubic equation.
