Search results
Results From The WOW.Com Content Network
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 to the computation of square and cube roots.
In the early 16th century, the Italian mathematician Scipione del Ferro (1465–1526) found a method for solving a class of cubic equations, namely those of the form x 3 + mx = n. In fact, all cubic equations can be reduced to this form if one allows m and n to be negative, but negative numbers were not known to him at that time. Del Ferro kept ...
The roots, stationary points, inflection point and concavity of a cubic polynomial x 3 − 6x 2 + 9x − 4 (solid black curve) and its first (dashed red) and second (dotted orange) derivatives. The critical points of a cubic function are its stationary points, that is the points where the slope of the function is zero. [2]
In algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x 3 = 2; in other words, x = , the cube root of two. This is because a cube of side length 1 has a volume of 1 3 = 1, and a cube of twice that volume (a volume of 2) has a side length of the cube root of 2.
More concretely, because the vector space of homogeneous polynomials P(x, y, z) of degree three in three variables x, y, z has dimension 10, the system of cubic curves passing through eight (different) points is parametrized by a vector space of dimension ≥ 2 (the vanishing of the polynomial at one point imposes a single linear condition).
The 1st equal areas cubic is the locus of a point X such that area of the cevian triangle of X equals the area of the cevian triangle of X*. Also, this cubic is the locus of X for which X* is on the line S*X, where S is the Steiner point. (S = X(99) in the Encyclopedia of Triangle Centers).
If the polynomial is irreducible and its coefficients are rational numbers (or belong to a number field), then the discriminant is a square of a rational number (or a number from the number field) if and only if the Galois group of the cubic equation is the cyclic group of order three.
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]