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As for the special case of a depressed cubic, this formula applies but is useless when the roots can be expressed without cube roots. In particular, if Δ 0 = Δ 1 = 0 , {\displaystyle \Delta _{0}=\Delta _{1}=0,} the formula gives that the three roots equal − b 3 a , {\displaystyle {\frac {-b}{3a}},} which means that the cubic polynomial can ...
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.
The polynomial P(x) has a rational root (this can be determined using the rational root theorem). The resolvent cubic R 3 (y) has a root of the form α 2, for some non-null rational number α (again, this can be determined using the rational root theorem). The number a 2 2 − 4a 0 is the square of a rational number and a 1 = 0. Indeed:
This already suffices to solve the quadratic by square roots. 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 ...
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.
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.
real root of x 3 = 2x ... ratio is found as the unique real solution of the cubic equation ... polynomial for the reciprocal root is the depressed cubic + ...