How do you solve a polynomial?

Question

like if you have

5 = 2x(cubed) + 6x(squared) + 8

how would you solve for x?

Answer 1

The polynomial you gave, namely:  2x^3 + 6x^2 + 3 = 0, is a cubic polynomial.  Using the rational roots theorem you can see that it has no rational roots, and it does not factor.  It has one real root whose approximate value is -3.151069.  You can find this using numerical methods, such as Newton's method.  See link below for that.  You can also, because it is a cubic with one real root, use Cardan's formula, discovered in the 1500's to write the exact solution in terms of square and cube roots.  That answer is hard to type in here, but click on this link and you will see it:

http://i33.tinypic.com/2uttc14.gif