Questions (I'd rather not make changes to the page since it's out of my area of expertise):
Further examples of polynomials (some are monomials which form a special case with only one term):
If somebody wants to integrate my writeup on E2 to here, feel free. The AC Method may be of particular interest. This is primarily just telling how to factor? polynomials so there might be a better place (i.e. factoring?) to put it. For simplicity, I'll post a partially wikified version here. If you think it's useful, integrate it. Else, just remove it: http://everything2.org/?node_id=895118 (Note: could contain some errors.)
anxn + an-1xn-1 + an-2xn-2. . . a1x + a0
The degree? of a polynomial is the highest total of powers? of variables? (x, y, etc.) of a single term?, so in the polynomial 2xy2 + x2 the degree is three (in the first term, x has a power of one). The standard form of a polynomial is when you write it with the degrees descending (x2 + x + 3, not x + x2 + 3)
To factor a polynomial (If you already know how to then skip down to the AC method. You'll like it. A lot.) you first factor out the [common factor]?, if there is one, using the [distributive property]?:
Ex 1) 2x^2 + 4x = 2x(x + 2)
With a binomial? (two terms, as in Ex 1) that's all. If you have a trinomial? (three terms, as in Ex 2) you're just getting started.
You usually have to find two binomials? (B1 and B2) whose first terms multiply to the first term of your trinomial, last terms multiply to the last term of the trinomial, and B1's first term times B2's last term plus vise versa equals the middle term (FOIL? users: Inside + Outside=Middle)
Ex 3) x2 + 3x + 2 = (x + 1)(x + 2)
If the first term of your trinomial has a coefficient? (a) of 1--as shown above--then the first terms of the binomials are x. Otherwise, you have to play around searching for the proper factors to get it right. That's where the following method comes in:
The AC Method
First factor out the [common factor]?. Always, always, always do this.
Ex 4) 6x2 + 2x-4
Now, I know you're thinking, "What if I have a four-term (or more) polynomial?" Easy: Take a few terms, and slap parenthesis around them (Hint, put together terms that have [common factors]? or that look like they'll factor easily.)
Ex 5) 2x3 - 3x2 + 4x - 6
That last example (first and last steps anyway) was taken from [College Algebra]? by [Michael Sullivan]? because I was having a heck of a time making up a good example. (I'm always coming up with prime? polynomials in my example and having to modify them so I can factor them. I wish my math teacher had let me do that in my homework.)
Now you need to do some heavy memorising. These are special polynomials and how to factor them. Knowing how to recognise them will help you enormously, both in multiplication? and factoring?:
[Difference of Squares]?: x2 - a2 = (x - a)(x + a) (Ex 6) x2 - 144 = (x + 12)(x - 12))
Take the coefficients? of (x + y)n and look at the nth row of [Pascal's Triangle]? (the "1" at the top is 0th). Cute and useful.
Sorry for the flood. :-)