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.
Another reliable source is Tony Hobson’s Just the Maths (individual pdfs hosted by UEA) (or a complete pdf from the Math Centre: AJ Hobson’s ‘Just the Maths’) includes a document on Polynomials: Just the Maths Algebra(8) – Polynomials.
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 .
Earlier I wrote about the excellent Helm Notes, this collection includes Solving Polynomial Equations which includes exercises with answers.
It is really useful to have a look at the graphical representation as well as the algebraic solution.
Show that x = −1 is a root of x3+11x2+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 Calculators is on Polynomials where you will find the following resources.
Division of Polynomials from Mathisfun – a really clear explanation:
Note the exercise at the end so you can practice, fully worked answers are shown.
WolframAlpha can of course handle polynomials.