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If only one root, say r 1, is real, then r 2 and r 3 are complex conjugates, which implies that r 2 – r 3 is a purely imaginary number, and thus that (r 2 – r 3) 2 is real and negative. On the other hand, r 1 – r 2 and r 1 – r 3 are complex conjugates, and their product is real and positive. [23]
Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: =. That difference is algebraically factorable as (+) (); if neither factor equals one, it is a proper factorization of N.
If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem. Singular cubic y 2 = x 2 ⋅ (x + 1). A parametrization is given by t ↦ (t 2 – 1, t ⋅ (t 2 – 1)).
d = 1: 2 and 1: two points determine a line, two lines intersect in a point, d = 2: 5 and 4: five points determine a conic, two conics intersect in four points, d = 3: 9 and 9: nine points determine a cubic, two cubics intersect in nine points, d = 4: 14 and 16. Thus these first agree for 3, and the number of intersections is larger when d > 3.
The Catalogue of Triangle Cubics is an online resource containing detailed information about more than 1200 cubic curves in the plane of a reference triangle. [1] The resource is maintained by Bernard Gibert. Each cubic in the resource is assigned a unique identification number of the form "Knnn" where "nnn" denotes three digits.
Iterating an infinite number of times results in the Jerusalem cube. Since the edge length of a cube of rank N is equal to that of 2 cubes of rank N+1 and a cube of rank N+2, it follows that the scaling factor must satisfy k 2 + 2 k = 1 {\displaystyle k^{2}+2k=1} , therefore k = 2 − 1 {\displaystyle k={\sqrt {2}}-1} which means the fractal ...
Many fixed-rate mortgages now offer lower interest rates than ARMs, with the average 30-year fixed mortgage rate at around 7%, compared to around 6.30% for a 5/1 ARM. It might be time to convert ...
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.