Polynomial Roots Calculator
Find every real root of a three-term polynomial a·xⁿ + b·xⁿ⁻¹ + c of degree 2, 3, or 4 (for example x² − 3x + 2, x³ − 2x² + 1 or x⁴ + x³ − 3). Lists all real roots, largest first, or reports that there are none.
Last updated: September 2026
Formula below · 1 source (Wikipedia) · Updated Sep 2026
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About this calculator
A root of a polynomial is a value of x that makes it equal zero. This calculator works with three-term polynomials p(x) = a·xⁿ + b·xⁿ⁻¹ + c, where n is the degree you choose (2, 3 or 4): a is the leading coefficient, b the coefficient of the next-highest power, and c the constant term. For n = 2 that is the full quadratic ax² + bx + c; for n = 3 it is ax³ + bx² + c (no x term); for n = 4 it is ax⁴ + bx³ + c (no x² or x term). Because p′(x) = xⁿ⁻²(n·a·x + (n − 1)·b), the polynomial can only turn around at x = 0 (for n ≥ 3) and at x = −(n − 1)b/(n·a). Between those turning points it is monotonic, so the calculator checks every interval out to the Cauchy bound |x| < 1 + max(|b/a|, |c/a|), brackets each sign change and bisects it to full precision; a turning point where p is zero is reported once as a repeated root. It returns every real root, largest first, or 'No real roots'. By the Fundamental Theorem of Algebra a degree-n polynomial has exactly n roots counted with multiplicity, including complex ones; complex roots are not listed here. Variables: degree selects n; coeff_a, coeff_b and coeff_c are a, b and c. Edge cases: a must be non-zero. For polynomials with more terms (a full cubic ax³ + bx² + cx + d, for example), use a general numerical root-finder (Newton-Raphson, Durand-Kerner) or a CAS such as SymPy.
How to use
Example 1 — Quadratic with two real roots. Find the roots of x² − 3x + 2 = 0. Enter Degree = 2, Leading = 1, Second = −3, Constant = 2. Result: x = 2, x = 1. ✓ Verify: (x − 1)(x − 2) = x² − 3x + 2, and the quadratic formula gives (3 ± √(9 − 8)) / 2 = 2 or 1. Example 2 — Cubic. Find the real roots of x³ − 2x² + 1 = 0. Enter Degree = 3, Leading = 1, Second = −2, Constant = 1. Result: x = 1.618034, x = 1, x = −0.618034. ✓ Verify: x = 1 gives 1 − 2 + 1 = 0; dividing out (x − 1) leaves x² − x − 1, whose roots are (1 ± √5)/2 ≈ 1.618034 and −0.618034. Example 3 — No real roots. For x² + 2x + 5 = 0 (Degree 2, Leading 1, Second 2, Constant 5) the discriminant is 4 − 20 = −16, so the calculator returns 'No real roots'; the complex roots are −1 ± 2i.
Frequently asked questions
How many roots does a polynomial have?
By the Fundamental Theorem of Algebra, a polynomial of degree n has exactly n roots, counted with multiplicity, when complex (non-real) roots are included. Real roots can be fewer: a quadratic has 0, 1 (a "double root"), or 2 real roots depending on the discriminant; a cubic always has at least 1 real root (because cubic polynomials with real coefficients are continuous and change sign); a quartic can have 0, 2, or 4 real roots. The remaining roots are complex and come in conjugate pairs for polynomials with real coefficients. This calculator lists every real root of its three-term polynomial; complex roots are not listed.
What is the quadratic discriminant and what does it tell you?
For ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. The sign of Δ determines the nature of the roots: Δ > 0 → two distinct real roots; Δ = 0 → one real root with multiplicity 2 (a double root, where the parabola touches the x-axis tangentially); Δ < 0 → two complex-conjugate roots (no real roots). The discriminant is also what appears under the square root in the quadratic formula, so its sign directly tells you whether the formula produces real or complex answers. A geometric interpretation: |Δ| / (4a²) is proportional to the squared distance between the parabola's vertex and the x-axis. The discriminant generalises: a cubic has its own discriminant (involving all four coefficients) that similarly distinguishes the cases of 1 vs 3 real roots.
Why doesn't this calculator handle the full quartic formula?
A general quartic ax⁴ + bx³ + cx² + dx + e = 0 has Ferrari's closed-form solution, but the formula is extraordinarily complex — it involves solving an intermediate cubic (the "resolvent cubic"), taking nested square roots, and handling multiple branches. Implementing it correctly requires complex-number arithmetic and careful case analysis. This calculator avoids Ferrari's formula: for its three-term quartic ax⁴ + bx³ + c it finds the turning points exactly and brackets each real root numerically, which is exact to the digits shown. For a general quartic with all five coefficients, use a numerical algorithm like Durand-Kerner or Jenkins-Traub (used by most modern CASes). The takeaway: this tool is exact for its three-term polynomials; for full cubics and quartics use a dedicated root-finder.
What are the most common mistakes people make finding polynomial roots?
The first is forgetting that polynomials can have complex roots — assuming "no real roots" means "no roots at all" leads to misinterpretation in fields like signal processing or control theory where the complex roots carry essential information. The second is plugging coefficients in the wrong order: for ax² + bx + c, a is the leading coefficient (xᵅ-coefficient), not the constant. The third is using a quadratic formula on a non-quadratic equation (cubic, exponential, transcendental); each polynomial degree has its own technique. The fourth is dividing the polynomial by the leading coefficient to "monic" form and then forgetting to restore the original scale — the roots are the same, but mis-handling can introduce confusion. The fifth is mistaking a multiplicity-2 root for two distinct roots; the quadratic formula collapses both branches to the same value when Δ = 0.
When should I not use this calculator?
Skip it for polynomials with more than three terms (a cubic with a non-zero x coefficient, or a quartic with a non-zero x² or x coefficient) — the three-coefficient form cannot represent them. It is the wrong tool for complex roots; this calculator lists real roots only and reports 'No real roots' when there are none. Avoid it for high-degree polynomials (degree 5+) where there is no general algebraic formula (Abel-Ruffini theorem) — numerical methods like Aberth, Jenkins-Traub, or eigenvalue-based companion-matrix methods are required. Finally, do not use it for polynomial factorisation in symbolic form; for rational-root or factor-theorem problems, use a polynomial-factoring calculator instead.