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In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series.It depends on the quantity | |, where are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one.
In mathematics, convergence tests are methods of testing for the convergence, ... The root test is stronger than the ratio test: whenever the ratio test determines ...
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.
The problem of counting and locating the real roots of a polynomial started to be systematically studied only in the beginning of the 19th century. In 1807, François Budan de Boislaurent discovered a method for extending Descartes' rule of signs —valid for the interval (0, +∞) —to any interval.
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
In mathematics, Descartes' rule of signs, described by René Descartes in his La Géométrie, counts the roots of a polynomial by examining sign changes in its coefficients. The number of positive real roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting zero coefficients), and the difference ...
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
According to the French school of mathematics of the 19th century, this is the first step in computing the real roots, the second being their approximation to any degree of accuracy; moreover, the focus is on the positive roots, because to isolate the negative roots of the polynomial p(x) replace x by −x (x ← −x) and repeat the process.
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