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Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots.
He understood the importance of the discriminant of the cubic equation to find algebraic solutions to certain types of cubic equations. [ 18 ] In his book Flos , Leonardo de Pisa, also known as Fibonacci (1170–1250), was able to closely approximate the positive solution to the cubic equation x 3 + 2 x 2 + 10 x = 20 .
In the case of a cubic equation, this resolvent is sometimes called the quadratic resolvent; its roots appear explicitly in the formulas for the roots of a cubic equation. The cubic resolvent of a quartic equation, which is a resolvent for the dihedral group of 8 elements.
In mathematics, a cubic function is a function of the form () = + + +, that is, a polynomial function of degree three. In many texts, the coefficients a , b , c , and d are supposed to be real numbers , and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to ...
In the case of a non-cyclic cubic field K this index formula can be combined with the conductor formula D = f 2 d to obtain a decomposition of the polynomial discriminant Δ = i(θ) 2 f 2 d into the square of the product i(θ)f and the discriminant d of the quadratic field k associated with the cubic field K, where d is squarefree up to a ...
The real part of the discriminant as a function of the square of the nome q on the unit disk. The modular discriminant Δ is defined as the discriminant of the characteristic polynomial of the differential equation ℘ ′ 2 ( z ) = 4 ℘ 3 ( z ) − g 2 ℘ ( z ) − g 3 {\displaystyle \wp '^{2}(z)=4\wp ^{3}(z)-g_{2}\wp (z)-g_{3}} as follows ...
In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is Graph of a polynomial function of degree 4, with its 4 roots and 3 critical points. + + + + = where a ≠ 0.
Graph of the polynomial function x 4 + x 3 – x 2 – 7x/4 – 1/2 (in green) together with the graph of its resolvent cubic R 4 (y) (in red). The roots of both polynomials are visible too. In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four: