# Polynomial

The Polynomials were a Greek speaking people living around 300BC-AD400 in the Peloponese. They founded their city state of Polynomiopolis on the Southern tip of the Penninsula in 288BC, having migrated after suffering persecution from Athenian Mathematicions who favoured the geometic approach rather than algebraic.

The Polynomial people were distinctive in their appearence, characterised by their finite sums of monomial terms and being differentiable at every point. In this respect they had much in common with the exponentials of Asia Minor, leading to the traditional view that their origins cam be found there.

This fragment of Polynomial pottery, found in 1986 shows a typical Polynomial trader:

$p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}$ Historians have been able to determine that the coeficients of p(x) were probably rational, and more so probably positive.

Archaologists excavating the area have been able to determine that during its peak Polynomiopolis probably looked something like:

Πολυνομιοπολις = F[x], where F = Some Field (or integral domain, or even just some ring!!!)

# Famous Polynomials

There has been one famous polynomial.

## Lagrange Polynomials

The Lagrange Polynomials were a philosophical cult that believed that the highest form of Polynomial was in fact an infinite "polynomial" which in fact isn't a polynomial at all. Instead this cult dressed in highly refined coeficients and were often mistaken for trignometric functions or exponentials.

# Religion

The Polynomials prided themselves on continuity and formal differentiability, these propeties took on a mystical significance which is reflected in their mythology.

## Polynomial Gods

The Polynomials believed in a pantheon of Gods.

### Zeus-inomial

The great king of the Polynomial Gods often depicted as:

Ζευσ-ινομιαλ = Ζ(x) = $987x^{556541}+654ix^{9723}+\pi ix^{59}$ , Ζ $\in C[x]$ ### Apollo-nomial

The God of integration and divisibility:

Απολλο-νομιαλ = Α(x) = ${\frac {5}{7}}x^{3}+{\frac {27}{7}}x^{2}+9x+1$ , Α $\in Q[x]$ ### Aphro-poly-dite

Goddess of love and binary opperations involving polynomials (i.e. multiplication and addition).

Αφρο-πολυ-διτε = Α(x) = ${\frac {5}{7}}x^{3}+{\frac {27}{7}}x^{2}+9x+1$ , Α $\in Q[x]$ ### MAthena

Goddess of abstract Polynomials, including formal differentiation.

ΜΑθηνα Μ(x) = $a_{234624}x^{234624}+a_{234623}x^{234623}...a_{1}x+a_{0}$ ### Hera (please insert maths related pun)

The Goddess of Polynomial solutions, Hera's anger at her husband and brother Zeus-inomial leads to a curse being placed on Polynomials of degree 5 or higher (c.f Galois 'kick in the balls' theory).

Ηερα = Η(x) = $a_{234624}x^{234624}+a_{234623}x^{234623}...a_{1}x+a_{0}$ ### Hephaesteonomial

God of numerical approximation. Hephaesteonomial is the metal worker of the Gods who uses brute force methods.

Ηφαιστος Η(x) = $33x^{2}-0.23$ ### Polyseidon

The God of the category of Polynomial rings.

Πολυσειδον = Π(x) = $3x^{3}-3x^{2}+3x-3$ The God of irreducible Polynomials with no roots.

Πολυηαδεσ = Π(x) = $4x^{4}-x^{2}+x$ ### Polynephone

The abducted wife of Polyhades.

Πολυνεφονη = Π(x) =$x^{3}-243x^{2}+226x-0.39881612114548984$ ## Mythology

The myths of the Polynomials are rich and include the Fraction War, the wanderings of Polydesseus, Jason and the Coeficients, the labours of Hilbertcles, Dividaeorlus and Integcus and King Euldipus.

# Properties

The polynomials were a deeply continuous people who practiced differentiation on a daily basis. However their society was deeply formal, and formal differentiation was practiced allowing Polynomials defined on arbitrary sets to flourish also. Common cultural practices included multiplication, addition, negation and differentiation.

## Taboos

A great many things were forbidden in Polynomial society, scholars today debate as to the origin/rational of these taboos.

### Root Extraction

A strongly taboo practice was root extraction, and many Polynomials clothed themselves in great numbers of factors to disguise their irreducible factors.

### Homomorphism

Homomorphism was a widely practiced though publicly condemned activity. Excavation evidence shows many polynomials were homomorphisms as this pottery shard show:

$f(x)=x^{2},f\in Z_{2}[x]$ f(x) is clearly a homomorphism as $f(xy)=x^{2}y^{2}=f(x)f(y)$ and $f(x+y)=(x+y)^{2}=x^{2}+y^{2}+2xy=f(x)+f(y)+2xy=f(x)+f(y)$ QED.

## 'fundamental theorem of Algebra'

The Polynomials believed that their definitions were the direct derivation of the 'fundamental theorem of Algebra' which stated that any nonconstant polynomial has a root. Historians however point out that Polynomiopolis was home to many for whom this law did not apply, leading experts to now think that the 'fundamental theorem of Algebra' is a direct derivation of the formulas of Analytolia