Polynomial roots

The calculator solves polynomial roots of any degree. For small degree polynomials analytic methods are applied, for 5-degree or higher the polynomial roots are estimated by numerical method.

The calculator solves real polynomial roots of any degree univariate polynomial with integer or rational terms. The calculator factors an input polynomial into several square-free polynomials, then solves each polynomial either analytically or numerically (for 5-degree or higher polynomials). A function graph is plotted to illustrate the polynomial solution.

PLANETCALC, N-degree polynomial roots

N-degree polynomial roots

Polynomial coefficients, space separated.
Digits after the decimal point: 5
Input polynomial
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N-degree polynomial real root calculation algorithm

  • Check whether the input polynomial even or odd - the polynomial is even if f(x) = f(-x), the polynomial is odd if f(x)=-f(-x).
  • Factor the polynomial into square-free polynomials with Yun algorithm Squarefree polynomial factorization.
  • Every n-degree polynomial obtained is solved analytically if n<5:
    • For 1st-degree - the root is the negative free term divided by the x coefficient
  • Use numeric methods If the polynomial degree is 5 or higher
    • Isolate the root bounds by VAS-CF algorithm: Polynomial root isolation. Find the roots in the positive field only if the input polynomial is even or odd (detected on 1st step)
    • For each isolation bound, find the approximate root value using the numeric method: Bisection method
    • Add the negative roots to the result set if the input polynomial is even or odd.
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PLANETCALC, Polynomial roots