Polynomial

A polynomial is a linear combination of powers of a variable (e.g., ax²+bx+c), supporting addition, subtraction, multiplication, division, differentiation, and integration; it is the foundation of algebra and numerical computing.

Purpose

To represent and manipulate algebraic expressions, the basis of interpolation, approximation, and equation solving.

Mechanism

P(x)=a_n×x^n+...+a_1×x+a_0. The degree is the highest power. Addition and subtraction work coefficient-wise; multiplication convolves coefficients. Evaluation uses Horner's method in O(n). Root finding: closed formulas for low degrees, Newton iteration for high degrees. FFT accelerates polynomial multiplication to O(n log n).

Use cases

Curve fitting and interpolation
Error-correcting codes (Reed-Solomon)
CRC generator polynomials
Numerical approximation

Examples

Input
```
2x²+3x+1 at x=2
```
Output
```
2×4+3×2+1=15
```
Horner evaluation

Common errors

High-degree polynomials are numerically unstable: Use orthogonal polynomials or center the variable

Related terms

A matrix is a rectangular array of numbers supporting addition, subtraction, multiplication, transposition, and inversion; it is the foundation of linear algebra and computer graphics.

A complex number has the form a+bi (i²=-1), extending the reals to two dimensions; it underpins signal processing and quantum computing.

BigInteger is an arbitrary-precision integer type, not limited by a fixed word size, used for computations beyond the native integer range (such as cryptography and combinatorics).

Related tools

Precision Calculator

High-precision arithmetic calculator. No floating-point errors. Free, private, 100% client-side.

References