For polynomial equations of degree < 5 see here.

An engineering problem (related to airplane engines) involves solving (with respect to x)
x^6 - 27*x^5 + 105*x^4 - 104*x^3 + 81*x^2 - 21*x + 2 = 0

We can draw the graph with JavaScript; for example by this script:

I understand that there is a solution between 0 and 10 and there is another between 10 and 25.
By using equp2.htm:

a = 0     b = 10     ...
a = 3.4443262222098245     b = 3.444326222209825
a = 10     b = 25     ...
a = 22.539168743888883     b = 22.539168743888887

The solutions are 3.44432622220982 and 22.53916874388888 (but we can be satisfied with a smaller number of digits).

We could do without drawing the graph. We could identify the initial intervals in which the solutions fall by using the [test] button.

[the problem is taken from "Science with Pocket Calculators" by David R. Green, John L. Lewis (1978)]

The script can also be used to tabulate polynomials. For example, in the case of the same function F considered above we have that F(3) = 533: