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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
Microsoft Math contains features that are designed to assist in solving mathematics, science, and tech-related problems, as well as to educate the user. The application features such tools as a graphing calculator and a unit converter. It also includes a triangle solver and an equation solver that provides step-by-step solutions to each problem.
the roots of this irreducible polynomial can be calculated as [5] 1 ± 2 1 / 6 , 1 ± − 1 ± 3 i 2 1 / 3 . {\displaystyle 1\pm 2^{1/6},1\pm {\frac {\sqrt {-1\pm {\sqrt {3}}i}}{2^{1/3}}}.} Even in the case of quartic polynomials , where there is an explicit formula for the roots, solving using the decomposition often gives a simpler form.
A root of a nonzero univariate polynomial P is a value a of x such that P(a) = 0. In other words, a root of P is a solution of the polynomial equation P(x) = 0 or a zero of the polynomial function defined by P. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.
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 cases where the function in question has multiple roots, it can be difficult to control, via choice of initialization, which root (if any) is identified by Newton's method. For example, the function f ( x ) = x ( x 2 − 1)( x − 3)e −( x − 1) 2 /2 has roots at −1, 0, 1, and 3. [ 18 ]
For t = −2 − 23 the first-order approximation 20 − t·20 19 /19! = 25.137... to the perturbed root 20.84... is terrible; this is even more obvious for the root α 19 where the perturbed root has a large imaginary part but the first-order approximation (and for that matter all higher-order approximations) are real.
A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction. For example, 3 is a square root of 9, since 3 2 = 9, and −3 is also a square root of 9, since (−3) 2 = 9.