On "polynomials"

**In mathematics (as we know from our university studies and from any reliable internet site, for example
Wolfram) ...**

A* function *F: x → *a _{n }*x

The coefficients can be constants, variables or more complex terms (as long as they do not contain x).

The

The

The word* polygon *derives from the Greek words *polís* (a lot) and *gonia* (angle), as a "figure with many angles".
Similarly the word* polynomial *derives from the Greek words *polís* and *ónoma* (name, expression), as "expression made up of the sum of many expressions".

We also speak of* trinomial*,

To talk about the degree of a polynomial* it is necessary to specify or that it is clear from the context what is the variable *to refer to.
For example A+A·B³ can be thought of as a polynomial (in particular, a binomial) of 3rd degree in B, but, transformed into A·(1+B³), it can also be considered a polynomial (in particular, a monomial) of 1st degree in A, or it can be considered as a polynomial of degree 0 in y: the variable y does not appear in it, therefore as regards y it is a constant. The variable against which a term is considered a polynomial is sometimes called **indeterminate**.

In the context of the study of two-input functions, the polynomials in 2 (or * more*)

It should be remembered that the concept of* function *must not be confused with that of* term*: the term x+7 is different from the function A → A+7 !

**When and why to study polynomials?**

In many textbooks the "polynomials" are studied by developing properties which are ** instead** properties of

But,

Because* at the end of the 2nd year of high school *(after starting the study of the functions in general) some features that* distinguish *them from the other functions can be brought into focus:

**• **because between the polynomials in an indeterminate it is possible to define, in analogy with the integers, a** division with remainder**;

**• **because the** remainder theorem **holds;

**• **because (as a consequence of it) we know that the** number of solutions **of a polynomial equation does not exceed its degree;

**• **and because with polynomial functions we can** approximate the other functions **(and "understand" how it is possible for a computer to calculate sine, exponential, …).

These are only the essential things related to the polynomials on which it makes sense to dwell (at the end of the second year and in the following years).

**Note**.
The polynomials* in 2 or more indeterminates *are univocally factorizable
[try to introduce in

**Something else
**

**Are there the greatest common divisor and the least common multiple between polynomials?**

The greatest common divisor between two positive integers is the greatest positivefor which both are divisible;
the greatest common divisor between two polynomials is integer a polynomial of maximumfor which both are divisible (or alternatively it is the degree monic polynomial - that is, the polynomial with directive coefficient 1 - of maximum degree for which both are divisible).For example 3·x²−6·x−9 [= 3·(x²−2·x−3) = 3·(x+1)·(x−3)] and 6·x²−6 [= 6·(x²−1) = 6·(x+1)·(x−1)] are both divisible by 3(x+1), but are also divisible by x+1, by 2x+2, by x/2+1/2, √3·x+√3, … If we want to choose a representative, we take the monic polynomial x+1. | |

f = function(x) f = function(x) 3*x^2-6*x-9; g = function(x) 6*x^2-6 plot(f,-2,2,col="blue") abline(h=0,v=0); abline(h=axTicks(2),v=axTicks(1),lty=3) plot(g,-2,2,col="red",add=TRUE); points(-1,0) |

Some textbooks teach pupils to also calculate the g.c.d. and l.c.m. of the highest degree coefficients! This is a typical example of a meaningless thing: not doing it* saves *time and does not cause* damage.*

* Two polynomials are "formally" equal if and only if they are "functionally" equal*. What does it mean? When does it make sense to address this question?

The meaning is this:
two polynomial functions of degrees n and m
*a _{n}*x

This

The principle (beyond the way it is expressed by textbooks) is completely incomprehensible to pupils. This is

Another

cos(x)² and 1−sin(x)² are formally different but are equal if you think of them as functions of x (the reason why things don't work in this case is that

In

0(+)0=0, 1(+)0=0(+)1=1, 1(+)1=0; 0(·)0=0, 1(·)1=1, 1(·)0=0(·)1=0

the polynomials x+1 and x³+1 are different while the functions x → x+1 and x → x³+1 coincide.

Without