Polynomial Roots
Find all real and complex roots of a polynomial up to degree 4 using exact algebraic methods.
About Polynomial Roots
This calculator uses exact algebraic methods for polynomials up to degree 4: the linear formula for degree 1, the quadratic formula for degree 2, Cardano's/trigonometric method for degree 3, and Ferrari's method for degree 4. All roots (real and complex) are reported.
Formulas
Quadratic formula
x = (-b ± √(b²-4ac)) / (2a)
- a
- Leading coefficient
- b
- Middle coefficient
- c
- Constant
Cubic discriminant test
Δ = q²/4 + p³/27
- p
- Depressed cubic linear term
- q
- Depressed cubic constant
Worked example
With the default values below, Polynomial Roots returns All Roots of 3, 1, 2. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Degreeone of 1 (Linear), 2 (Quadratic), 3 (Cubic), 4 (Quartic)
- 3 (Cubic)
- a (leading coefficient)
- 1
- b
- -6
- c
- 11
- d
- -6
- e
- 0
Results
- All Roots
- 3, 1, 2
- Real Roots
- 3
- Complex Roots
- 0
Step by step
- 1Values used
- 2Quadratic formula
- 3Cubic discriminant test
- 4All Roots= 3, 1, 2
- 5Real Roots= 3
- 6Complex Roots= 0
Assumptions and limits
- The result is derived only from the 6 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
Why is the maximum degree 4?
There is no general closed-form solution for polynomials of degree 5 or higher (Abel-Ruffini theorem). Degrees 1-4 have exact algebraic formulas.
What does it mean when a root is complex?
Complex roots appear as a±bi where i=√(-1). They always come in conjugate pairs for polynomials with real coefficients.