Polynomial

In algebra, a polynomial is, roughly speaking, a formal expression obtained from constant numbers and one or several unspecified numbers called "variables", denoted by letters like $x$, $y$, etc., by making a finite number of additions, subtractions and multiplications. For instance, $x^{2}-2x+1$ is a polynomial of one variable, $x$, whereas $x^{2}+y^{2}$ is a polynomial of two variables, $x$ and $y$. Expressions like ${\frac {x-1}{x^{2}+2}}$ or ${\sqrt {x^{2}+1}}$ are not polynomials ; the first one is a rational function, and the second one is an irrational expression, due to the square root symbol. Such operations might be expressed within the constant numbers, as in the example ${\frac {1}{2}}x^{3}+x-{\sqrt {2}}$, but this is only because ${\frac {1}{2}}$ and ${\sqrt {2}}$ are elements of the set (e.g. real numbers) that are being used as coefficients of the polynomials.

It may be convenient to think of a polynomial as a function of its variables, that is, $x\mapsto x^{2}-2x+1$ or $(x,y)\mapsto x^{2}+y^{2}$. Such a function is called a polynomial function. But in reality, both concepts are different, the unspecified variables being purely formal entities when one thinks of an abstract polynomial, whereas they are meant to be replaced by any number when one thinks of a function. The distinction is important in abstract algebra, because what we have called "constant numbers" is more generally replaced by any ring, and for some rings the two concepts cannot be identified. There is not such a problem with polynomials over rings of usual numbers like integers, rational, real or complex numbers. Still it is important to understand that calculations with polynomials can be conceived in an only formal way, without giving any special ontological status to the variables. To make the distinction clear, it is common in algebra to denote the abstract variables with capital letters ($X$, $Y$, etc.), while variables of functions are still denoted with lowercase letters. We will use this convention in what follows.

Polynomials of one variable

In this section we deal with the simplest case, that is, polynomials of only one variable, denoted $x$. A polynomial can be written as a finite sum of terms, called monomials. Each monomial is either a constant, or a constant times a positive whole number power of x. For instance, 1, $2x^{4}$, and $-3x^{2}$ are monomials, and their sum, $2x^{4}-3x^{2}+1$ is a polynomial.

A coefficient equal to 1 in front of a positive power of x is typically dropped from the notation, so that $x^{2}-x+1$ represents the same polynomial as $1x^{2}-1x+1$. It is sometimes useful to explicitly write a power of x in each monomial, even the constants. To accomplish this, you can write $x^{0}$ after the constant, so that 2 and $2x^{0}$ are considered the same.

Degree

The power of the variable appearing in a monomial is the degree of the monomial. By the above convention, a constant c is the same as $cx^{0}$ and has degree equal to 0. The degree of a polynomial is the largest of the degrees of the monomials appearing in the polynomial. The only exception is the constant polynomial 0, which typically is not assigned a degree (for reasons made clear below). As an example, 2 has degree 0, $x^{2}+x$ has degree 2, and $-3x^{4}+x^{2}+x^{5}$ has degree 5.

The degree is an important identifier when working with polynomials. For instance, many procedures for factoring or solving polynomial equations require identifying the degree of the polynomial first. In the last example above, we had to scan through the polynomial from the left all the way through the right to determine that the degree is 5. To facilitate identifying the degree of a polynomial, as well as manipulations of polynomials, they are usually written in standard form. The standard form of a polynomial is obtained by combining terms of the same degree, and then writing the monomials so that the exponents decrease from left to right. The degree of a polynomial in standard form is the degree of the first monomial appearing. The term of highest degree is the leading term and its coefficient is the leading coefficient. A polynomial with leading coefficient equal to 1 is monic. We can put the last example above in standard form by rearranging the monomials to obtain $x^{5}-3x^{4}+x^{2}$. It is of course just as easy to work with polynomials where the monomials are written so that the degrees increase from left to right.

Definition

Let us consider some expressions like $X^{2}-2X+1$, ${\frac {1}{2}}X^{3}+X-{\sqrt {2}}$, or $2X^{5}-3X^{2}+1$. We can write all of them as follows:

$X^{2}-2X+1=1+(-2)X+1X^{2}+0X^{3}+0X^{4}+\cdots ,$

${\frac {1}{2}}X^{3}+X-{\sqrt {2}}=-{\sqrt {2}}+1X+0X^{2}+{\frac {1}{2}}X^{3}+0X^{4}+\cdots ,$

$2X^{5}-3X^{2}+1=1+0X+(-3)X^{2}+0X^{3}+0X^{4}+2X^{5}+0X^{6}+\cdots .$

This suggests that a polynomial can be entirely defined by giving a sequence of numbers, which are called its coefficients, all of them being zero from some rank. For instance the three polynomials above can be written respectively $(1,-2,1,0,0,\cdots )$, $\left(-{\sqrt {2}},1,0,{\frac {1}{2}},0,\cdots \right)$, and $(1,0,-3,0,0,2,0,\cdots )$, the dots meaning the sequence continues with an infinity of zeros. This leads to the definition below.

Definition. A polynomial $P$, over the ring $R$ is a sequence $P=\left(a_{0},a_{1},a_{2},\cdots ,a_{n},\cdots \right)$ of elements of $R$, called the coefficients of $P$, this sequence containing only a finite number of nonzero terms. The rank of the last nonzero term is called the degree of the polynomial.

Hence, the degrees of the three polynomials given above are respectively 2, 3 and 5. By convention, the degree of $(0,0,\cdots )$ is set to $-\infty$.

This definition may surprise the reader, because in reality, one thinks of a polynomial as an expression of the form $a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}$ rather than $\left(a_{0},a_{1},a_{2},\cdots ,a_{n},\cdots \right)$. We will progressively show how to return to this usual way of writing a polynomial. First, we identify any element $a_{0}$ of the ring to the polynomial $\left(a_{0},0,0,\cdots \right)$. For instance, we write only $7$ instead of the cumbersome $\left(7,0,0,\cdots \right)$, (or in the familiar fashion $7+0X+0X^{2}+\cdots$).

Secondly, we merely denote by $X$ the polynomial

$X=\left(0,1,0,0,\cdots \right)$.

This is natural, as in the familiar fashion this sequence corresponds to $0+1X+0X^{2}+0X^{3}+\cdots$ It remains to give a sense to $X^{2}$, $X^{3}$, etc. This will be made in the next two subsections.

Calculation rules

We now define addition and multiplication of polynomials, beginning with addition, which is easy.

Addition

With the traditional notation, if we have $P=2X^{5}-3X^{2}+1$ and $Q=-X^{5}+4X^{4}+2X^{2}-1$, we want to have $P+Q=(2-1)X^{5}+4X^{4}+(-3+2)X^{2}+1-1=X^{5}+4X^{4}-X^{2}$, that is, one wants to add coefficients separately for each degree. This leads to the formal definition below.

Definition. Given two polynomials $P=\left(a_{0},a_{1},a_{2},\dots \right)$ and $Q=\left(b_{0},b_{1},b_{2},\dots \right)$, the sum $P+Q$ is defined by $P+Q=\left(a_{0}+b_{0},a_{1}+b_{1},a_{2}+b_{2},\dots \right)$.

Multiplication

Multiplication is harder to define. Let us begin with an example using traditional notation. For $P=X^{2}+X-2$ and $Q=2X^{2}-3X+1$, we want to have

$PQ=X^{2}\left(2X^{2}-3X+1\right)+X\left(2X^{2}-3X+1\right)-2\left(2X^{2}-3X+1\right)$; $PQ=2X^{4}+(-3+2)X^{3}+(1-3-2\cdot 2)X^{2}+(1-2\cdot (-3))X-2$; $PQ=2X^{4}-X^{3}-6X^{2}+7X-2\$.

One can observe that the coefficient of say, $X^{2}$, is obtained by adding $1\cdot 1$, $1\cdot (-3)$ and $-2\cdot 2$, that is, by adding all the $a_{i}b_{j}$ so that $i+j=2$, where the $a_{i}$ denote the coefficients of $P$ and the $b_{j}$ those of $Q$. Those mechanics lead to give the definition below.

Definition. Given two polynomials $P=\left(a_{0},a_{1},a_{2},\cdots \right)$ and $Q=\left(b_{0},b_{1},b_{2},\cdots \right)$, the product $PQ$ is defined by $PQ=\left(c_{0},c_{1},c_{2},\cdots \right)$, where for every index $k$, the coefficient $c_{k}$ is given by $c_{k}=\sum _{i+j=k}a_{i}b_{j}$.

The reader which is upset by those cumbersome notations should just retain that this definition allows to multiply polynomials (considered as mere sequences of coefficients) as one is used to do in elementary algebra (using the traditional notation, as in the example). The only striking fact is that in our construction, $X$ does not represent a number, but a pure abstract entity for which we have defined some rules of calculation.

The algebra $R[X]$

With the definition above, one can verify that the product of the polynomial $X=\left(0,1,0,0,\dots \right)$ by itself, that is $X^{2}$, is the sequence $X^{2}=\left(0,0,1,0,0,\dots \right)$. More generally, for each natural number $n$, one can verify that the $n$-th power of $X$ is given by $X^{n}=\left(0,\dots ,0,1,0,0,\dots \right)$, where the $1$ is the coefficient of index $n$ and all other coefficients are zeros. In particular, we have the usual convention $X^{0}=\left(1,0,0,\dots \right)$, which we identified to the constant $1$.

Now, any polynomial $P=\left(a_{0},a_{1},a_{2},\dots ,a_{n},0,0,\dots \right)$ is exactly equal to $a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}$, where the addition and the powers (which are mere repetitions of multiplications) are defined as in the preceding subsections. Our whole construction legitimates the traditional notation, and from now on, we will only use the later, with which calculations use natural rules of elementary algebra. It is however important to remember that the "variable" $X$ did not denote some number in our construction, but a particular sequence of coefficients. We have succeeded in defining polynomials in a purely formal manner.