Polynomial equations are mathematical expressions set equal to zero, with solutions known as roots. This overview covers solving techniques from algebraic methods for lower-degree polynomials to numerical methods for higher-degree ones. It also touches on the Fundamental Theorem of Algebra, which guarantees at least one complex root for non-constant polynomials, and explores the Abel-Ruffini theorem's implications on solvability by radicals.
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Polynomial equations are expressions that equate a polynomial to zero
Terms
Terms are the building blocks of polynomial equations, consisting of a coefficient and a variable raised to a non-negative integer exponent
Standard Form
The standard form of a polynomial equation in one variable is \(a_n x^n + a_{n-1}x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0 = 0\), where \(a_n, a_{n-1}, \dots, a_1, a_0\) are constants, and \(x\) represents the variable
The solutions or roots of a polynomial equation are the values of the variable that satisfy the equation and make it equal to zero
Algebraic methods such as factoring, completing the square, and using the quadratic formula can be used to solve linear and quadratic equations, while numerical methods are often used for higher-degree equations
The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root, and the number of times a polynomial will equal zero is equal to its degree
The roots of a polynomial are the values of the variable that make the polynomial equal to zero
The multiplicity of a root is the number of times it is repeated as a factor of the polynomial
Vieta's formulas establish a relationship between the coefficients of a polynomial and the sums and products of its roots
The Abel-Ruffini theorem and Galois theory show that there is no general algebraic solution for polynomial equations of degree five or higher
Newton's method, the Durand-Kerner method, and other root-finding algorithms are used to approximate solutions for higher-degree polynomial equations
In the case of polynomials with several variables, algebraic geometry uses techniques such as Gröbner bases to find solutions or prove that no solutions exist
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Solving linear and quadratic equations
Use algebraic methods like factoring, completing the square, or quadratic formula.
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Explicit formulas for cubic and quartic equations
Cubic and quartic equations have complex explicit formulas, unlike linear and quadratic.
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Abel-Ruffini theorem on polynomial equations
No general algebraic solution for polynomial equations of degree five or higher.
