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The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is a n = 1.
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.
For finding one root, Newton's method and other general iterative methods work generally well. For finding all the roots, arguably the most reliable method is the Francis QR algorithm computing the eigenvalues of the companion matrix corresponding to the polynomial, implemented as the standard method [1] in MATLAB.
In mathematics, exponentiation, denoted b n, is an operation involving two numbers: the base, b, and the exponent or power, n. [1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, b n is the product of multiplying n bases: [1] = ⏟.
The rule states that if the nonzero terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). [1] These operations may be performed on numbers, in which case they are often called arithmetic operations.
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).