Polynomial calculator

Evaluate, differentiate, integrate, and find real roots of any polynomial — plus add or multiply two polynomials. Enter coefficients comma-separated, highest-degree first.

Polynomial P(x) coefficients (highest degree first)

P(x) = x³ − 6x² + 11x − 6

Polynomial Q(x) — for add / multiply only

Q(x) = x − 1

Point x = — for evaluation

Real roots of P(x)

x = 1, 2, 3

P(x₀)

P′(x)

3x² − 12x + 11

∫P(x) dx

0.25x⁴ − 2x³ + 5.5x² − 6x + C

P(x) + Q(x)

x³ − 6x² + 12x − 7

P(x) × Q(x)

x⁴ − 7x³ + 17x² − 17x + 6

How real roots are found

For degree 1 and 2, exact closed-form solutions are used. For degree 3 and 4, the calculator combines closed-form Cardano / Ferrari with numeric refinement. For higher degrees, it uses Durand-Kerner iteration, then filters real roots within a tolerance.

Complex roots (where the imaginary part is non-trivial) are not displayed. Use a CAS like SymPy or Wolfram for symbolic complex roots.

Coefficient input

Enter coefficients with the highest degree first. Examples:

1, -6, 11, -6 → x³ − 6x² + 11x − 6
2, 0, -8 → 2x² − 8
1, 0, 0, 0, -1 → x⁴ − 1

Polynomial calculator

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How real roots are found

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