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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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## Definition of Polynomial Equations

### Polynomial equations are mathematical statements that set a polynomial equal to zero

Polynomial equations are expressions that equate a polynomial to zero

### Components of Polynomial Equations

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

### Solutions and Roots of Polynomial Equations

The solutions or roots of a polynomial equation are the values of the variable that satisfy the equation and make it equal to zero

## Solving Polynomial Equations and the Fundamental Theorem of Algebra

### Methods for Solving Polynomial Equations

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

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

## Roots of Polynomials and Their Properties

### Definition of Roots

The roots of a polynomial are the values of the variable that make the polynomial equal to zero

### Multiplicity of Roots

The multiplicity of a root is the number of times it is repeated as a factor of the polynomial

### Vieta's Formulas

Vieta's formulas establish a relationship between the coefficients of a polynomial and the sums and products of its roots

## Approaches to Solving Higher-Degree Polynomial Equations

### Limitations of Solving Higher-Degree Polynomial Equations

The Abel-Ruffini theorem and Galois theory show that there is no general algebraic solution for polynomial equations of degree five or higher

### Numerical Methods for Solving Higher-Degree Polynomial Equations

Newton's method, the Durand-Kerner method, and other root-finding algorithms are used to approximate solutions for higher-degree polynomial equations

### Algebraic Geometry

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