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Lagrange's method can be applied directly to the general cubic equation ax 3 + bx 2 + cx + d = 0, but the computation is simpler with the depressed cubic equation, t 3 + pt + q = 0. Lagrange's main idea was to work with the discrete Fourier transform of the roots instead of with the roots themselves.
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry . The theory is simplified by working in projective space rather than affine space , and so cubic surfaces are generally considered in projective 3-space P 3 ...
For example, finding a substitution = + + for a cubic equation of degree =, = + + + such that substituting = yields a new equation ′ = + ′ + ′ + ′ such that ′ =, ′ =, or both. More generally, it may be defined conveniently by means of field theory , as the transformation on minimal polynomials implied by a different choice of ...
Tschirnhausen cubic, case of a = 1. In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation = where sec is the secant function.
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 ...
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubic equation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} applied to homogeneous coordinates ( x : y : z ) {\displaystyle (x:y:z)} for the projective plane ; or the inhomogeneous version for the affine space determined by setting z = 1 in such an ...
Degree 3. Cubic plane curves include Cubic parabola; ... An elementary treatise on cubic and quartic curves by Alfred Barnard Basset (1901) online at Google Books
A counterexample by Ernst S. Selmer shows that the Hasse–Minkowski theorem cannot be extended to forms of degree 3: The cubic equation 3x 3 + 4y 3 + 5z 3 = 0 has a solution in real numbers, and in all p-adic fields, but it has no nontrivial solution in which x, y, and z are all rational numbers. [1]