What does it mean to find the roots of a polynomial?

The roots (or zeros) of a polynomial are the values of x where the polynomial equals zero. For example, the roots of x² − 5x + 6 are 2 and 3, because substituting either value gives zero. Roots may be real numbers or complex (involving the imaginary unit i).

Which formulas are used?

For degree 2: the standard quadratic formula. For degree 3: Cardano's formula (with the trigonometric method for three distinct real roots). For degree 4: Ferrari's resolvent cubic method. All computations are performed in the browser - no server required.

What are complex roots?

When the discriminant is negative, some roots involve the square root of a negative number. These are expressed as a + bi, where a is the real part and b is the imaginary part. Complex roots of polynomials with real coefficients always come in conjugate pairs (a + bi and a − bi).

Why can't I solve degree 5 or higher?

The Abel–Ruffini theorem proves there is no general algebraic formula (using radicals) for polynomials of degree 5 or higher. Numerical methods (like Newton's method) can approximate roots to arbitrary precision, but exact closed-form solutions do not exist in general.

How do I enter coefficients?

Enter the coefficients from highest to lowest degree. For ax² + bx + c, enter a, b, c. For x³ − 6x² + 11x − 6, enter 1, −6, 11, −6. Terms with zero coefficients must still be included (e.g., x³ + 0x² + x − 2 is entered as 1, 0, 1, −2).

Polynomial Solver - Roots of Degree 2, 3, and 4

Polynomial curve
x	x^2 − 5x + 6
-0.5000	8.7500
0.1667	5.1944
0.8333	2.5278
1.5000	0.7500
2.1667	-0.1389
2.8333	-0.1389
3.5000	0.7500
4.1667	2.5278
4.8333	5.1944
5.5000	8.7500

Polynomial curve

x^2 − 5x + 6

-0.5000

8.7500

0.1667

5.1944

0.8333

2.5278

1.5000

0.7500

2.1667

-0.1389

2.8333

-0.1389

3.5000

0.7500

4.1667

2.5278

4.8333

5.1944

5.5000

8.7500

What is a polynomial equation?

A polynomial equation is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀ = 0, where the aᵢ are real-number coefficients. The degree is the highest power of x with a non-zero coefficient. Solving the equation means finding all values of x (called roots or zeros) that satisfy it.

Quadratic (degree 2)

For ax² + bx + c = 0 the quadratic formula gives roots x = (−b ± √(b² − 4ac)) / (2a). The term under the square root is the discriminant: positive -> two distinct real roots; zero -> one repeated real root; negative -> two complex conjugate roots.

Cubic (degree 3)

Cardano's formula (1545) provides an exact algebraic solution for all cubics. When the discriminant is negative, there are three distinct real roots; the tool uses the trigonometric (cosine) form to avoid the "casus irreducibilis" issue with cube roots of complex numbers.

Quartic (degree 4)

Ferrari's method reduces a quartic to a resolvent cubic, whose roots are used to split the quartic into two quadratics. Each quadratic is then solved with the quadratic formula, yielding all four roots.

Why degree 5+ is different

The Abel–Ruffini theorem (1824) proves that polynomials of degree 5 or higher have no general algebraic solution in terms of radicals. Galois theory later explained this in terms of the symmetry groups of the roots. Numerical solvers (Newton's method, Durand–Kerner, etc.) can approximate roots to any desired precision.

Worked examples

Degree	Equation	Discriminant	Roots
2	x² − 5x + 6 = 0	25 − 24 = 1 > 0	x = 2, x = 3
2	x² − 4x + 4 = 0	16 − 16 = 0	x = 2 (repeated)
3	x³ − 6x² + 11x − 6 = 0	Δ > 0 (3 real)	x = 1, 2, 3
4	x&sup4; − 5x² + 4 = 0	—	x = ±1, ±2

Complex root visualization

For polynomials with real coefficients, complex roots always appear in conjugate pairs a ± bi. This is a consequence of the complex conjugate root theorem. Geometrically, when the discriminant is negative the parabola (degree 2) does not cross the x-axis — the two complex conjugate roots correspond to where the parabola “would” cross if extended into the complex plane.

Numerical methods for degree 5+

Newton’s method starts from an initial guess x⊂0 and iterates x⊂n+1 = x⊂n − f(x⊂n)/f′(x⊂n) until convergence. It converges quadratically near a simple root but can diverge for poor starting guesses.

The Durand–Kerner algorithm simultaneously refines all n roots at once, making it robust for finding all roots of a degree-n polynomial. Both methods return approximate rather than exact answers because no algebraic closed form exists for degree ≥ 5.