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

Approximation Numerical

(Usually represented by the symbol ≈) is an inexact representation of something that is still close enough to be useful. Although approximation is most often applied to numbers, it is also frequently applied to such things as mathematical functions, shapes, and physical laws.

Approximations may be used because incomplete information prevents use of exact representations. Many problems in physics are either too complex to solve analytically, or impossible to solve using the available analytical tools. Thus, even when the exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly.

For instance, physicists often approximate the shape of the Earth as a sphere even though more accurate representations are possible, because many physical behaviours—e.g. gravity—are much easier to calculate for a sphere than for other shapes.

It is difficult to exactly analyze the motion of several planets orbiting a star, for example, due to the complex interactions of the planets' gravitational effects on each other, so an approximate solution is effected by performing iterations. In the first iteration, the planets' gravitational interactions are ignored, and the star is assumed to be fixed. If a more precise solution is desired, another iteration is then performed, using the positions and motions of the planets as identified in the first iteration, but adding a first-order gravity interaction from each planet on the others. This process may be repeated until a satisfactorily precise solution is obtained. The use of perturbations to correct for the errors can yield more accurate solutions. Simulations of the motions of the planets and the star also yields more accurate solutions.

As another example, in order to accelerate the convergence rate of evolutionary algorithms, fitness approximation - that leads to build model of the fitness function to choose smart search steps - is a good solution.

The type of approximation used depends on the available information, the degree of accuracy required, the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation.

A brief definition of what a mathematical model

A mathematical model uses mathematical language to describe a system. The process of developing a mathematical model is termedmathematical modelling(also modeling). Mathematical models are used not only in the natural sciences (such as physics, biology, earth,metereology) and engineering disciplines, but also in the social sciences (such as economics, psycology ,sociology and political science); physicists engineers,computer scientists, and economist use mathematical models most extensively.

Eykhoff (1974) defined a mathematical model as a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form.

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models,differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures.