Talk: Polynomial

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Early usage: For LarrySanger

At the high school level quadratic equations are useful in displaying the teacher's facility in proving the quadratic formula, by completing the square. ---- No thanks necessary.----

Questions (I'd rather not make changes to the page since it's out of my area of expertise):

Can I have more examples please? See below.
What are polynomials good for? Describing some kinds of relations between variables. Like much of math, we present definitions and techniques and leave the applications to others.
Maybe an example of a high school textbook problem involving polynomials would help orient me...just a suggestion, feel totally free to ignore it.

Further examples of polynomials (some are monomials which form a special case with only one term):

Area of a square = side²
Volume of a cube = side³.
Area of a square with its lower left corner at the point (x,x) and its upper right corner at (y,y) = (y-x)² = y²-2xy+x².

This is a polynomial in the two variables, x and y.

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.)

a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-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)

Ex 2) 2x² + 6x + 8 = 2(x² + 3x + 4)

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.

Now you have ax² + bx + c a isn't 1.
Change it to x² + bx + ac. (If you're stuck wondering how the hell to move the a all the way over to the c, don't bother. Just do it.)
Factor x² + bx + ac into your (presumably) two binomials. Then stick a back into the first terms of both of them, factor out the common factor and toss it out. You're done.

Ex 4) 6x² + 2x-4

2(3x² + x-2) (Factor out common factor)
2(x² + x-6) (move a to third term)
2(x + 3)(x-2) (factor)
2(3x + 3)(3x-2) (put a back into first terms)
2(x + 1)(3x-2) (factor out and delete common factor)
If you're planning on using the AC Method a lot you may want to work on your factoring large numbers because ac is often rather large.

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

(2x³ - 3x²) + (4x - 6)
x²(2x - 3) + 2(2x - 3)
(x² + 2)(2x - 3)

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))

Perfect Squares: x² ± 2ax + a² = (x ± a)²
Unnamed, but bears remembering: x² (a + b)x + ab = (x + a)(x + b)
Unnamed, but bears remembering: acx² + (ad + bc)x + bd = (ax + b)(cx + d)
Perfect Cubes: x³ + 3ax² + 3a²x + a³ = (a + x)³, x³ - 3ax² + 3a²x - a³ = (a - x)³
Sum of Two Cubes: x³ + a³ = (x + a)(x² - ax + a²)
Difference of Two Cubes: x³ - a³ = (x - a)(x² + ax +a²)

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. :-)

If this flood might be useful to someone, maybe it belongs on related corrolary pages. stevertigo

Technical point: I've always seen a polynomial defined as an expression, not an equation or a function, ie anx^n + ... + a0. The term "polynomial" is later loosely applied to graphs, functions and equations with a polynomial. -- user:Tarquin

Just wanted to draw everyone's attention to the fact that an anonymous user just changed "In algebra" to "In calculus". with so many mathematicians at Wikipedia, I find it difficult to believe that such an elementary mistake exists in a basic article, and it sounds like something a semi-educated person might think is true. Personally, I haven't the foggiest notion of what calculus is, much less if... calculators? (calculites? calculians?) use polynomials or not. Tokerboy 03:01 Nov 22, 2002 (UTC)

See calculus :-) -- Tarquin

Algebra is a subject. Calculus, on the other hand, is something of a hodge-podge --- a collection of subjects that the curriculum brings together. Algebra goes far beyond those things that most students see, and is a subject to which careers of some researchers are devoted. The topics that go far beyond calculus, on the other hand, are not called "calculus", but go by other names, such as "analysis" and "topology". Therefore, it makes sense to say "in algebra", but not as much to say "in calculus". Polynomials of course appear in calculus, as do many things from algebra. -- Mike Hardy

The reason I separated calculus and algebra is that in algebra, one has to distinguish between polynomials and polynomial functions, while in calculus one doesn't. This point is now lost, in fact the first sentence seems to suggest that the two concepts are the same, which they are only in sloppy calculus usage. AxelBoldt 23:41 Nov 30, 2002 (UTC)

I think this article would be improved if some knowledgeable person would add a few sentences about the Fuchsian Function solution to the paragraph which discusses roots of nth order polynomials. They are hinted at with the existing phrase "degree 5 eluded researchers for a long time", which suggests that a solution was eventually found, but this solution is not mentioned in the article. A new article on Fuchsian Funtions would also be welcome. kielhorn@portland.quik.com Dec 22, 2002

Does anyone know anything about "polynomial arithmetic modulo 2" - you know, the mathematics used for cyclic redundancy checks? Because I don't, and it's not explained in the CRC article, either. -- Tim Starling

Removing this:

Polynomials are important because they are the simplest functions: their definition involves only addition and multiplication (since the powers are just shorthands for repeated multiplications).

Polynomials are surely not the "simplest"; surely f(x) = 0 is "simpler". In addition, something being simple does not imply that it is important. This sentence contributes nothing of value. -Ryguasu 21:11, 13 Sep 2003 (UTC)

Contents

1 Degrees past the 4th

2 Complexity

3 Restructuring

4 Definiton and history

5 Scary definition

Degrees past the 4th

So far, the polynomial page has special names for degrees up to the 4th. How about past 4th?? Do any of these make sense?? Degree Names from 1 to 12

Linear
Quadratic
Cubic
Quartic
Quintic
Sextic
Septic
Octic
Nonic
Decic
Unidecic
Duodecic

66.32.148.219 00:54, 10 Apr 2004 (UTC)

Possibly, but people don't use "special" names for high-degree polynomials. Compare with n-gon. People just say a nth degree polynomial, or the polynomial has degree n. Dysprosia 00:57, 10 Apr 2004 (UTC)

The furthest I've heard is "quintic". -- Tarquin 08:08, 10 Apr 2004 (UTC)

Sextic, yes. I'd be careful from there on. It may be octavic, for example, for degree 8 sometimes. Septimic, too. Charles Matthews 08:24, 10 Apr 2004 (UTC)

Complexity

This paragraph seems a bit confused - is it talking about computational complexity, or bounding general polynomials in magnitude by their leading term?

Charles Matthews 08:24, 14 Jul 2004 (UTC)

I moved the Complexity paragraph to big o notation as an example. MathMartin 16:05, 11 Nov 2004 (UTC)

Restructuring

I think the article is in horrible shape. I have rewritten the definition and restructured the existing material. Some key points of the article should be

history (finding roots, galois theory)
numerical analysis
abstract algebra

MathMartin 21:47, 16 Aug 2004 (UTC)

Nice work! :) I agree that overview and history should come first: they are the parts the layperson can understand.-- Tarquin 21:56, 16 Aug 2004 (UTC)

Definiton and history

I think the definition of a (mathematical) term should always be the first subsection of its entry. There must also be an Analysis section under mathematics to which the polynomials entry should be moved. Who's responsible for that?

It works fine as it is. There may be a way of casually introducing what a polynomial is, before the formal definition sections, however. Dysprosia 01:56, 24 Oct 2004 (UTC)

Scary definition

I do not like the latest edits on the definition. Polynomials are a basic topic which should to accessible to a wide range of people. But now the definition will scare off even undergraduate math students. We should have a simple definition which covers the most common cases and terms and then later in the article we can always add the scary stuff for the fearless.MathMartin 16:12, 11 Nov 2004 (UTC)

Yes, the discussion doesn't cut it. Charles Matthews 22:33, 11 Nov 2004 (UTC)

I'll third that. Paul August 22:47, Nov 11, 2004 (UTC)

I reverted the definition to the more simple one I wrote some time ago. Perhaps someone else can integrate the more abstract definition below into the article. MathMartin 13:43, 17 Nov 2004 (UTC)

Let us first note that in most cases the term polynomial refers to a term of the following form:

<math>\sum_{i=1}^n{a_i x^i},\qquad n\in\mathbf{N},a_i\in\mathbf{Q}.<math>

However, this use is, although common, somewhat unprecise since what is actually meant is an univariate polynomial over Q according to the general definition:

Let r, s and t be elements of N, x₁,...,x_r be variables, F a field and

<math>M=\{\prod_{j=0}^r x_j^{k_j}: k_j\in\mathbf{N}\land 0\le k_j\le s\}.<math>

Furthermore let the finitely many elements of M be denoted by y₁,...,y_t. Then an r-variate polynomial over F is a term of the form

<math>\sum_{i=1}^t{a_i y_i},\qquad a_i\in\it{F}.<math>

The a_i are called coefficients. The coefficient of

is called constant coefficient. Polynomials with only one, two or three non-zero coefficients are called monomials, binomials and trinomials, respectively.

The term polynomial can also refer to a function p: M->N, x->p(x), where p(x) is a polynomial as defined above. Such a function may also be called a polynomial function.

The leading coefficient of an univariate polynomial is the coefficient a_k which doesn't equal 0 and also has a_i=0 for all i > k. We say that a univariate polynomial has degree (or order) k if a_k is its leading coefficient, and we say that it is monic or normed if its leading coefficient is 1.

Univariate polynomials of

degree 0 are called constant functions,
degree 1 are called linear functions,
degree 2 are called quadratic functions,
degree 3 are called cubic functions,
degree 4 are called quartic functions and
degree 5 are called quintic functions.

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