Solvability by Radicals: When Polynomial Equations Have Algebraic Solutions
Explore when polynomial roots can be expressed using finite arithmetic and root extractions—solvability by radicals, classic formulas for quadratics/cubics/quartics, Abel–Ruffini limits, and Galois theory criteria.
An algebraic solution is an explicit formula for one or more roots of an algebraic equation that can be expressed using a finite combination of the basic arithmetic operations and root extractions. In other words, the values of the unknown(s) are written in terms of the equation's coefficients by applying addition, subtraction, multiplication, division and taking nth roots (square roots, cube roots, etc.). Such formulas are often described as being “solvable by radicals.”
Classic examples
The best-known closed-form result is the formula for the general quadratic equation. The solutions of
for the quadratic
(with a ≠ 0) are expressed in terms of the coefficients a, b and c by arithmetic and a square root.
There also exist algebraic formulas for the general cubic and the general quartic, though these expressions are considerably more involved than the quadratic formula. These higher-degree formulas still use only the permitted operations and radicals.
Limits: why not all polynomials have algebraic solutions
The Abel–Ruffini theorem establishes that for a general polynomial of degree five or higher there is no solution expressible solely by the listed operations and a finite number of root extractions. That is, a generic quintic cannot be solved by radicals. Modern understanding of when a polynomial admits an algebraic solution comes from Galois theory, which gives criteria (the structure of the polynomial's Galois group) determining whether its roots are obtainable by radicals.
Special cases
Even though the general polynomial of degree ≥ 5 is not solvable by radicals, many particular higher-degree equations are. For example, the simple equation
has the algebraic solution
More generally, whether a particular polynomial admits an algebraic formula depends on its symmetry properties rather than only on its degree.
Operations used in algebraic solutions
- Addition
- Subtraction
- Multiplication
- Division
- Extraction of roots (taking square roots, cube roots, etc.)
Questions and answers
Q: What is an algebraic solution?
A: An algebraic solution is an algebraic expression which is the solution of an algebraic equation in terms of the coefficients of the variables. It can be found using addition, subtraction, multiplication, division, and extraction of roots (square roots, cube roots, etc.).
Q: What is a well-known example of an algebraic solution?
A: The most well-known example is the solution of the general quadratic equation.
Q: Is there a more complicated solution for higher degree equations?
A: Yes, there is a more complicated solution for the general cubic equation and quartic equation.
Q: Does every polynomial equation have an algebraic solution?
A: No, according to Abel-Ruffini theorem states that the general quintic equation does not have an algebraic solution. This means that the general polynomial equation of degree n, for n ≥ 5 cannot be solved by using only algebra.
Q: Are there any conditions under which we can get an algebraic solutions for higher degree equations?
A: Yes, under certain conditions we can get an algebraic solutions; for example, the equation x^10 = a can be solved as x = a^(1/10).
Q: How do you solve a quadratic equation?
A: To solve a quadratic equation you need to use addition, subtraction multiplication and division as well as extracting square roots or other types of roots from it.
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Author
AlegsaOnline.com Solvability by Radicals: When Polynomial Equations Have Algebraic Solutions Leandro Alegsa
URL: https://en.alegsaonline.com/art/2532