Real Roots

Some articles on real roots, root, real root, real, roots:

Casus Irreducibilis - Relation To Angle Trisection
... the reducible and irreducible cubic cases with three real roots is related to the issue of whether or not an angle with rational cosine or rational sine is trisectible by the classical means of compass ... of this angle has a cosine that is one of the three real roots of the equation Likewise, if the sine of is known to have a particular rational value, then one third of this angle has a sine that is one of ... On the other hand, if the rational root test shows that there is no real root, then casus irreducibilis applies, or is not constructible, the angle is not constructible, and the angle is not ...
Greatest Common Divisor Of Two Polynomials - Univariate Polynomials With Coefficients in A Field - GCD and Root Finding - Sturm Sequence
... The Sturm sequence of a polynomial with real coefficients is the sequence of the remainders provided by a variant of Euclid's algorithm applied to the polynomial and its derivative ... Sturm's theorem asserts that V(a)-V(b) is the number of real roots of the polynomial in the interval ... Thus the Sturm sequence allows to compute the number of real roots in a given interval ...
Roots of A Cubic Function - Trigonometric (and Hyperbolic) Method
... When a cubic equation has three real roots, the formulas expressing these roots in terms of radicals involve complex numbers ... been proved that when none of the three real roots is rational—the casus irreducibilis— one cannot express the roots in terms of real radicals ... Nevertheless, purely real expressions of the solutions may be obtained using hypergeometric functions, or more elementarily in terms of trigonometric functions ...
Sturm's Theorem
... Sturm's theorem expresses the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values of the Sturm's sequence at the ... Applied to the interval of all the real numbers, it gives the total number of real roots of p ... Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, Sturm's theorem yields the number of distinct real roots and locates them in intervals ...

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