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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.
Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filippa Ferro.His father, Floriano, worked in the paper industry, which owed its existence to the invention of the press in the 1450s and which probably allowed Scipione to access various works during the early stages of his life.
In the case of the cubic, Tschirnhaus transformations replace the variable by a quadratic function, thereby making it possible to eliminate two terms, and so can be used to eliminate the linear term in a depressed cubic to achieve the solution of the cubic by a combination of square and cube roots.
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 van der Waals equation of state may be written as (+) =where is the absolute temperature, is the pressure, is the molar volume and is the universal gas constant.Note that = /, where is the volume, and = /, where is the number of moles, is the number of particles, and is the Avogadro constant.
which is a depressed quartic equation. If b = 0 {\displaystyle \ b=0\ } then we have the special case of a biquadratic equation , which is easily solved, as explained above. Note that the general solution, given below, will not work for the special case b = 0 . {\displaystyle \ b=0\ .}
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.
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