Roots of a higher order polynomial

Question

I tried solving $4z^4-8z^3+z^2-2z+1=0$ by myself then failed and used Wolfram for finding roots. Now my question is: can the roots of a polynomial with degree of at least $4$ be found with some tools rather than curve sketching or using a calculator?

Answer 1

For quartic polynomials, there are Ferrari's method and Descartes' method. Above degree $4$, there is no algebraic method, according to the Abel-Ruffini theorem.

Answer 2

You can sketch the function by finding the roots of the derivative and computing the locations of the extrema. From there, draw and estimate the values of the roots.

This is a recursive procedure.

But you can't escape evaluating the polynomial(s) at some points, which requires some... stuff.