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Using the formula relating the general cubic and the associated depressed cubic, this implies that the discriminant of the general cubic can be written as (+). It follows that one of these two discriminants is zero if and only if the other is also zero, and, if the coefficients are real , the two discriminants have the same sign.
There are conjectures about whether del Ferro worked on a solution to the cubic equation as a result of Luca Pacioli's short tenure at the University of Bologna in 1501–1502. Pacioli had previously declared in Summa de arithmetica that he believed a solution to the equation to be impossible, fueling wide interest in the mathematical community.
In some cases, the concept of resolvent cubic is defined only when P(x) is a quartic in depressed form—that is, when a 3 = 0. Note that the fourth and fifth definitions below also make sense and that the relationship between these resolvent cubics and P ( x ) are still valid if the characteristic of k is equal to 2 .
For a general formula that is always true, one thus needs to choose a root of the cubic equation such that m ≠ 0. This is always possible except for the depressed equation y 4 = 0. Now, if m is a root of the cubic equation such that m ≠ 0, equation becomes
In the special case of a depressed cubic polynomial + +, the discriminant simplifies to − 4 p 3 − 27 q 2 . {\displaystyle -4p^{3}-27q^{2}\,.} The discriminant is zero if and only if at least two roots are equal.
The cubic-plus-chain (CPC) [28] [29] [30] equation of state hybridizes the classical cubic equation of state with the SAFT chain term. [21] [22] The addition of the chain term allows the model to be capable of capturing the physics of both short-chain and long-chain non-associating components ranging from alkanes to polymers. The CPC monomer ...
So, if the three non-monic coefficients of the depressed quartic equation, + + + =, in terms of the five coefficients of the general quartic equation are given as follows: =, = + and = +, then the criteria to identify a priori each case of quartic equations with multiple roots and their respective solutions are exposed below.
The cubic virial equation of state at is: = (+ +) It can be rearranged as: (+ +) = The factor / is the volume of saturated gas according to the ideal gas law, and can be given a unique name : = In the saturation region, the cubic equation has three roots, and can be written alternatively as: () = which can be expanded as: (+ +) + (+ +) = is a ...