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.

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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 polynomial, 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
Real roots
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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 numeric method: Bisection method
    • Add the negative roots to the result set if the input polynomial is even or odd.