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If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.
In particular, when the number of sign changes is zero or one, then there are exactly zero or one positive roots. A linear fractional transformation of the variable makes it possible to use the rule of signs to count roots in any interval. This is the basic idea of Budan's theorem and the Budan–Fourier theorem. Repeated division of an ...
[29] [30] His solution of the quadratic equation + = was as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is ...
In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968. The algorithm transforms the problem of integration into a problem in algebra.
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator.
Root-finding of polynomials – Algorithms for finding zeros of polynomials; Square-free polynomial – Polynomial with no repeated root; Vieta's formulas – Relating coefficients and roots of a polynomial; Cohn's theorem relating the roots of a self-inversive polynomial with the roots of the reciprocal polynomial of its derivative.
CLT – central limit theorem. cod, codom – codomain. cok, coker – cokernel. colsp – column space of a matrix. conv – convex hull of a set. Cor – corollary. corr – correlation. cos – cosine function. cosec – cosecant function. (Also written as csc.) cosech – hyperbolic cosecant function. (Also written as csch.) cosh ...
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields.