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The characteristic equation of a third-order constant coefficients or Cauchy–Euler (equidimensional variable coefficients) linear differential equation or difference equation is a cubic equation. Intersection points of cubic Bézier curve and straight line can be computed using direct cubic equation representing Bézier curve.
In 1936 Margharita P. Beloch showed that use of the 'Beloch fold', later used in the sixth of the Huzita–Hatori axioms, allowed the general cubic equation to be solved using origami. [1] In 1949, R C Yeates' book "Geometric Methods" described three allowed constructions corresponding to the first, second, and fifth of the Huzita–Hatori axioms.
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In all, Cardano was driven to the study of thirteen different types of cubic equations (chapters XI–XXIII). In Ars Magna the concept of multiple root appears for the first time (chapter I). The first example that Cardano provides of a polynomial equation with multiple roots is x 3 = 12 x + 16, of which −2 is a double root.
Casus irreducibilis (from Latin 'the irreducible case') is the name given by mathematicians of the 16th century to cubic equations that cannot be solved in terms of real radicals, that is to those equations such that the computation of the solutions cannot be reduced the the computation of square and cube roots.
y = x 3 for values of 1 ≤ x ≤ 25.. In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 2 3 = 8 or (x + 1) 3.
Cubic equations, which are polynomial equations of the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is a rational number).
Folds satisfying the Huzita–Hatori axioms can construct exactly the same set of points as the extended constructions using a compass and conic drawing tool. Therefore, origami can also be used to solve cubic equations (and hence quartic equations), and thus solve two of the classical problems. [19]