Polynomial Root Calculator
Polynomial Root Calculator
Enter Values
How to use this calculator
- Enter each coefficient of your polynomial into the matching field (x⁴, x³, x², x and the constant). Leave any term you don't have blank — a blank field counts as zero.
- The degree is detected automatically from the highest non-zero coefficient you enter, so the same tool handles quadratics, cubics and quartics.
- Read the degree and the list of real roots. Expand the working panel to see the discriminant and the method used.
How it works
Worked example
Solve x³ − 6x² + 11x − 6 = 0. Enter 1 as the x³ coefficient, −6 as x², 11 as x and −6 as the constant. The calculator detects a degree-3 (cubic) polynomial and returns three real roots: x₁ = 1, x₂ = 2 and x₃ = 3 — because x³ − 6x² + 11x − 6 factors as (x − 1)(x − 2)(x − 3).
Common mistakes
- Skipping a missing term instead of leaving it blank. In x³ − 7x + 6 the x² term is zero, so leave the x² field empty (it means 0) rather than typing the next coefficient in the wrong box.
- Expecting complex roots. This tool lists real roots only — a quadratic like x² + x + 1 correctly reports 'No real roots' because its solutions are complex.
- Putting the constant in the leading field. The highest-power coefficient sets the degree, so entering everything shifted by one place changes which equation you're solving.
Frequently asked questions
What degree of polynomial can this solve?
Degrees 2, 3 and 4 (quadratic, cubic and quartic), plus a simple linear case. It uses exact closed-form formulas — the quadratic formula, Cardano's method for cubics and Ferrari's method for quartics — rather than approximate numerical root-finding.
Why does it say 'No real roots'?
Some polynomials only cross zero at complex values. For example x² + x + 1 has a negative discriminant, so both of its roots are complex numbers. This calculator reports real roots only, so it shows 'No real roots' in that case.
How do I enter a polynomial with a missing term?
Leave that coefficient's field blank — a blank counts as zero. To solve x³ − 7x + 6 = 0, enter 1 for x³, leave x² empty, enter −7 for x and 6 for the constant. The roots are −3, 1 and 2.
Does it show repeated roots?
A repeated (double or triple) root is listed once. The degree still tells you the total number of roots counting multiplicity, so a cubic with only two distinct real roots has one of them repeated.
Related tools
- Quadratic Equation Calculator
- Nth Root Calculator
- GCD & LCM Calculator
- Linear Equation Solver
- Graphing Calculator
- System of Equations Calculator
Explore more in Maths, Education & Science.
Tip: Enter any known values to calculate the remaining results.
All calculations run in your browser. Your inputs are never saved or transmitted.