p(x)Polynomial Root Calculator
e.g. '1 -6 11 -6' = x³-6x²+11x-6
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).
- 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]
x³−6x²+11x−6=0 (coefficients: 1 −6 11 −6)=x=1, x=2, x=3(x−1)(x−2)(x−3)=0
x²+1=0 (coefficients: 1 0 1)=No real roots — complex roots ±i
| Degree | Name | Max Real Roots | Formula |
|---|---|---|---|
| 1 | Linear | 1 | x = −b/a |
| 2 | Quadratic | 2 | Quadratic formula |
| 3 | Cubic | 3 | Cardano's formula (1545) |
| 4 | Quartic | 4 | Ferrari's formula (1540s) |
| 5+ | Quintic / higher | n | No general algebraic solution (Abel-Ruffini 1824) |
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