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James V. Uspensky gave a bound on the length of the continued fraction (the integer k in Vincent's theorem), for getting zero or one sign variations: [1] [7] Theorem (Uspensky) — Let p ( x ) be a polynomial of degree n , and sep( p ) be the minimal distance between two roots of p .
When dealing with the polynomial p(x) in one variable, one defines the number of sign variations of p(x) as the number of sign variations in the sequence of its coefficients. Two versions of this theorem are presented: the continued fractions version due to Vincent, [1] [2] [3] and the bisection version due to Alesina and Galuzzi. [4] [5]
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 numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function f is a number x such that f ( x ) = 0 . As, generally, the zeros of a function cannot be computed exactly nor expressed in closed form , root-finding algorithms provide approximations to zeros.
The goal of these steps is usually to isolate the variable one is interested in on one side, a process known as solving the equation for that variable. For example, the equation x − 7 = 4 {\displaystyle x-7=4} can be solved for x {\displaystyle x} by adding 7 to both sides, which isolates x {\displaystyle x} on the left side and results in ...
The rational univariate representation or RUR is a representation of the solutions of a zero-dimensional polynomial system over the rational numbers which has been introduced by F. Rouillier. [10] A RUR of a zero-dimensional system consists in a linear combination x 0 of the variables, called separating variable, and a system of equations [11]
Finding the root of a linear polynomial (degree one) is easy and needs only one division: the general equation + = has solution = /. For quadratic polynomials (degree two), the quadratic formula produces a solution, but its numerical evaluation may require some care for ensuring numerical stability.
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.
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