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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.
Its true value is the real solution of the equation x 3 = 2x ... the reciprocal root is the depressed cubic ... the Binet formula for the related sequence = ...
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.
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
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.
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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The graph of any cubic function is similar to such a curve. The graph of a cubic function is a cubic curve, though many cubic curves are not graphs of functions. Although cubic functions depend on four parameters, their graph can have only very few shapes. In fact, the graph of a cubic function is always similar to the graph of a function of ...