Finding the roots of polynomials is a long-standing problem that has been extensively studied throughout the history and substantially influenced the development of mathematics. It involves determining either a numerical approximation or a closed-form expression of the roots of a univariate polynomial, i.e., determining approximate or closed form solutions of ${\displaystyle x}$ in the equation

${\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}=0}$

where ${\displaystyle a_{i}}$ are either real or complex numbers.

Efforts to understand and solve polynomial equations led to the development of important mathematical concepts, including irrational and complex numbers, as well as foundational structures in modern algebra such as fields, rings, and groups.

Despite being historically important, finding the roots of higher degree polynomials no longer play a central role in mathematics and computational mathematics, with one major exception in computer algebra.

Closed-form formulas for polynomial roots exist only when the degree of the polynomial is less than 5. The quadratic formula has been known since antiquity, and the cubic and quartic formulas were discovered in full generality during the 16th century.

When the degree of polynomial is at least 5, a closed-form expression for the roots by the polynomial coefficients does not exist in general, if we only uses additions, subtractions, multiplications, divisions, and radicals (taking n-th roots) in the formula. This is due to the celebrated Abel-Ruffini theorem. On the other hand, the fundamental theorem of algebra shows that all nonconstant polynomials have at least one root. Therefore, root-finding algorithms consists of finding numerical solutions in most cases.

Root-finding algorithms can be broadly categorized according to the goal of the computation. Some methods aim to find a single root, while others are designed to find all complex roots at once. In certain cases, the objective may be to find roots within a specific region of the complex plane. It is often desirable and even necessary to select algorithms specific to the computational task due to efficiency and accuracy reasons. See Root Finding Methods for a summary of the existing methods available in each case.