How to find the roots of a septic polynomial?

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

A septic polynomial is a polynomial of degree 7. It can be difficult to find the roots of a septic polynomial algebraically, so numerical methods such as Newton-Raphson or bisection can be used. These methods involve making an initial guess for a root and then using iterative calculations to refine the estimate until a satisfactory level of accuracy is reached.

Alternatively, if the septic polynomial can be factorised, the roots can be found by setting each factor equal to zero and solving for the roots. For example, if the septic polynomial is (x-1)(x+2)(x-3)(x+4)(x-5)(x+6)(x-7), the roots are x=1, -2, 3, -4, 5, -6, and 7.

If the septic polynomial cannot be factorised, we can use the rational root theorem to find possible rational roots. This involves finding all the factors of the constant term and all the factors of the leading coefficient, and then testing all possible combinations of these factors as potential rational roots. Once a rational root is found, we can use polynomial long division to reduce the polynomial to a lower degree and continue the process until all roots are found.

In summary, finding the roots of a septic polynomial can be challenging, but numerical methods and factorisation can be used to find the roots. If factorisation is not possible, the rational root theorem can be used to find possible rational roots.