Polynomial Roots Calculator
Find every real and complex root of a polynomial.
3 real, 0 complex
| # | Root | Type |
|---|---|---|
| 1 | 1 | real |
| 2 | 3 | real |
| 3 | 2 | real |
A polynomial’s roots are the x-values that make it zero, and a degree-n polynomial has exactly n of them (counting complex and repeated roots). For x³ − 6x² + 11x − 6, the rational root theorem flags ±1, ±2, ±3, ±6 — testing shows the roots are 1, 2, and 3, so it factors as (x − 1)(x − 2)(x − 3).
Solving any polynomial
The quadratic formula handles degree 2, but higher-degree polynomials need a numerical approach. This tool uses the Durand–Kerner method, which refines guesses for all roots simultaneously until they converge — finding real and complex roots together. To hunt for roots by hand, the rational root theorem narrows the candidates first.
the rational root theorem: a shortlist to test for nice integer or fraction roots
Worked example
Find the roots of x³ − 6x² + 11x − 6 = 0:
- 1 List rational candidates. Constant 6, leading coefficient 1, so candidates are ±1, ±2, ±3, ±6.
- 2 Test them. x = 1: 1 − 6 + 11 − 6 = 0 ✓. x = 2: 8 − 24 + 22 − 6 = 0 ✓. x = 3: 27 − 54 + 33 − 6 = 0 ✓.
- 3 Confirm the factorisation. Three roots for a cubic: (x − 1)(x − 2)(x − 3) = x³ − 6x² + 11x − 6. The roots are 1, 2, 3.
How many roots, by degree
By the fundamental theorem of algebra, a degree-n polynomial has exactly n roots, counting multiplicity, over the complex numbers.
| Degree | Total roots | Notes |
|---|---|---|
| 1 (linear) | 1 | Always one real root |
| 2 (quadratic) | 2 | Real or a conjugate pair |
| 3 (cubic) | 3 | At least one real root |
| 4 (quartic) | 4 | Complex roots come in pairs |
| n | n | Real roots + conjugate complex pairs |
Real and complex roots
Complex roots come in conjugate pairs. For a real-coefficient polynomial, if a + bi is a root then so is a − bi. That is why an odd-degree polynomial always has at least one real root — the complex ones pair off.
Multiplicity counts. A repeated root, such as (x − 2)², counts twice toward the total of n. The graph touches the x-axis there instead of crossing it.
The results are precise estimates. Durand–Kerner returns high-precision numerical roots; for clean polynomials they land on the exact integers or rationals you would find by hand.