Polynomial roots

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.

N-degree polynomial roots

Polynomial

Polynomial coefficients, space separated.

Show graph

Calculation precision

Digits after the decimal point: 5

Calculate

Input polynomial

 

Even

 

Odd

 

Factorization

 

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Graph

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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
• • 2nd degree is solved by Solution of quadratic equation
• • 3rd degree: Cubic equation
• • 4th degree: Quartic equation solution
• 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.