How to solve an nth degree polynomial equation

The typical approach of solving a quadratic equation is to solve for the roots

x=b±b24ac2a

Here, the degree of x is given to be 2

However, I was wondering on how to solve an equation if the degree of x is given to be n.

For example, consider this equation:

a0xn+a1xn1++an=0

Answer

There is no perfect answer to this question. For polynomials up to degree 4, there are explicit solution formulas similar to that for the quadratic equation (the Cardano formulas for third-degree equations, see here, and the Ferrari formula for degree 4, see here).

For higher degrees, no general formula exists (or more precisely, no formula in terms of addition, subtraction, multiplication, division, arbitrary constants and n-th roots). This result is proved in Galois theory and is known as the Abel-Ruffini theorem. Edit: Note that for some special cases (e.g., xna), solution formulas exist, but they do not generalize to all polynomials. In fact, it is known that only a very small part of polynomials of degree 5 admit a solution formula using the operations listed above.

Nevertheless, finding solutions to polynomial formulas is quite easy using numerical methods, e.g., Newton’s method. These methods are independent of the degree of the polynomial.

Attribution
Source : Link , Question Author : Ayush Khemka , Answer Author : Johannes Kloos

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