For notes and examples on polynomials, see the following resources: From the mathcentre which has an extensive collection of very clear notes and other resources, the Algebra collection includes Polynomial Division.

David Smith’s site ‘The Maths Teacher’ includes Algebraic Multiplication and Division under Algebra AS level.

In Paul’s Online Notes, Section 1-4 is on Polynomials; these start with a clear definition and include examples of polynomials and also expressions which are not polynomials .

It is really useful to have a look at the graphical representation as well as the algebraic solution. For example: Show that x = −1 is a root of x^{3}+11x^{2}+31x+21 = 0 and locate the other roots algebraically. The graph on Desmos is here. You can check your algebraic solution, by looking at the graph.

One of the pages in the series on Calculatorsis on Polynomials where you will find the following resources.