The roots of a polynomial are the places where the polynomial evaluates to 0 (zero). In other words, if one were to graph a polynomial, the roots would be all the places where the function crosses the x axis. For example, consider the function x + 1. To solve for the root of x + 1, set the whole function to 0:

0 = x + 1

and solve for x (x should evaluate to -1). With polynomials of greater degree, it gets a little more complicated. Here is x² - 4, which is a polynomial of degree 2:

0 = x² - 4

There is no easy way to solve this, but luckily, x² - 4 factors into:

0 = (x+ 2)(x - 2)

and we find that the roots are 2 and -2 (plug each of these in for x and solve the equation and you'll see why). A polynomial of degree n has at most n roots in the real number system and always has exactly n roots in the complex number system. It has been proven that polynomials of a degree greater than 4 cannot be solved, but functions such as Newton's method can estimate roots to them.

Care should be taken with finite fields, as factoring a polynomial means finding factors like (x-a) and factors like (x^2+1) over Z_3.

I mean that if you check that a polynomial doesn't evaluate as 0 on any x, it doesn't mean it hasn't got any factors.

Bernoulli's Method For the Solution of Polynomial Equations

A number analytic means of solving polynomial equations, equations of the form
a₀xⁿ + a₁x^n-1 + ... = 0
(Where a_i is a constant and may be complex).

In its simplest form, given here, it finds the root of maximum modulus.

First form the corresponding difference equation:
s_n = (a₁s_n-1 - ...)/a₀.
Then use the implied recurrence relation to find the sequence s₀, s₁, etc..

The sequence of quotients q₀, q₁, etc. then converges to the root of maximum modulus. Where q_n = s_n+1/s_n.

Simple With An Example

Take the equation x² - x - 1 = 0.

The associated difference equation is: s_n = s_n-1 + s_n-2

Take the first two elements of the sequence S₀, s₁, ... to be 0, 1. Then {s_n} = 0, 1, 1, 2, 3, 5, 8, 13, 21, .... (The famous Fibonacci - the way rabbit populations allegedly grow - sequence).

The sequence of quotients is 1/0, 1/1, 2/1, 3/2, 5/3, .... This converges to (1+sqrt5)/2 - a solution of the original equation and the Greeks' "Golden" ratio.