Solving Polynomial Equations

In algebra you spend lots of time solving polynomial equations or factoring polynomials (which is the same thing). It would be easy to get lost in all the techniques, but this paper ties them all together in a coherent whole.

Factor = Root

Make sure you aren’t confused by the terminology. All of these are the same:

• Solving a polynomial equation p(x) = 0
• Finding roots of a polynomial equation p(x) = 0
• Finding zeroes of a polynomial function p(x)
• Factoring a polynomial function p(x)

There’s a factor for every root, and vice versa. (x−r) is a factor if and only if r is a root. This is theFactor Theorem: finding the roots or finding the factors is essentially the same thing. (The main difference is how you treat a constant factor.)

Exact or Approximate?

Most often when we talk about solving an equation or factoring a polynomial, we mean an exact (or analytic) solution. The other type, approximate (or numeric) solution, is always possible and sometimes is the only possibility.

When you can find it, an exact solution is better. You can always find a numerical approximation to an exact solution, but going the other way is much more difficult. This page spends most of its time on methods for exact solutions, but also tells you what to do when analytic methods fail.

For more information please go to http://oakroadsystems.com/math/polysol.htm