How to find the roots of a sextic polynomial?

To find the roots of a sextic polynomial, we can use numerical methods or factorisation.

One numerical method is the Newton-Raphson method, which involves iteratively improving an initial guess for a root. Another method is the Bairstow's method, which is a modified version of the Newton-Raphson method that can handle complex roots.

Factorisation can also be used to find the roots of a sextic polynomial. One approach is to use the Rational Root Theorem to identify possible rational roots, and then use polynomial long division to factorise the polynomial. Another approach is to use the Factor Theorem to factorise the polynomial into quadratic factors, and then solve for the roots of each quadratic.

Alternatively, we can use the cubic formula or the quartic formula to reduce the sextic polynomial to a cubic or quartic polynomial, respectively, and then use numerical methods or factorisation to find the roots of the reduced polynomial. However, these formulas can be quite complicated and may not always be practical to use.

Overall, finding the roots of a sextic polynomial can be a challenging task, and it may require a combination of numerical methods and factorisation techniques.