Polynomial Roots
Find all real and complex roots of a polynomial up to degree 4 using exact algebraic methods.
Inputs
For degree ≥ 2
For degree ≥ 3
For degree 4 only
All Roots
3, 1, 2
Real Roots
3
Complex Roots
0
Step by step
Values used
Degree = 3 (Cubic); a (leading coefficient) = 1; b = -6; c = 11; d = -6; e = 0
Quadratic formula
x = (-b ± √(b²-4ac)) / (2a)
Cubic discriminant test
Δ = q²/4 + p³/27
All Roots
= 3, 1, 2
Real Roots
= 3
Complex Roots
= 0
How it works
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
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.