How to Calculate Polynomial Root

What is Polynomial Root?

A polynomial root (or zero) is a value of x where the polynomial equals zero: p(x) = 0. Finding roots is fundamental in algebra, engineering, physics, and numerical methods. The Fundamental Theorem of Algebra guarantees that a degree-n polynomial has exactly n roots (counting complex roots and multiplicities).

Formula

Degree 1 (linear): ax+b=0 → x = −b/a

b: 0 → x = −b/a — 0 → x = −b/a
x: (−b ± √(b²−4ac)) / 2a — (−b ± √(b²−4ac)) / 2a

Step-by-Step Guide

1Degree 1 (linear): ax+b=0 → x = −b/a
2Degree 2 (quadratic): ax²+bx+c=0 → x = (−b ± √(b²−4ac)) / 2a
3Degree 3–4: closed-form formulas exist (complex, rarely used)
4Degree 5+: no general closed-form solution (Abel-Ruffini theorem)
5Our calculator uses bisection search for real roots in [−20, 20]

Worked Examples

Input

x³−6x²+11x−6=0 (coefficients: 1 −6 11 −6)

Result

x=1, x=2, x=3

(x−1)(x−2)(x−3)=0

Input

x²+1=0 (coefficients: 1 0 1)

Result

No real roots — complex roots ±i

Frequently Asked Questions

What is Polynomial Roots?

A polynomial root (or zero) is a value of x where the polynomial equals zero: p(x) = 0. Finding roots is fundamental in algebra, engineering, physics, and numerical methods

How accurate is the Polynomial Roots calculator?

The calculator uses the standard published formula for polynomial roots. Results are accurate to the precision of the inputs you provide. For financial, medical, or legal decisions, always verify with a qualified professional.

What units does the Polynomial Roots calculator use?

This calculator works with inches. You can enter values in the units shown — the calculator handles all conversions internally.

What formula does the Polynomial Roots calculator use?

The core formula is: Degree 1 (linear): ax+b=0 → x = −b/a. Each step in the calculation is shown so you can verify the result manually.

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