Polynomials/Field/Introduction/Section

Definition

Let ${}K$ be a field. An expression of the form

${}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,,$

with ${}a_{i}\in K$ and ${}n\in \mathbb {N}$,

is called a polynomial in one variable over ${}K$.

The numbers ${}a_{0},a_{1},\ldots ,a_{n}$ are called the coefficients of the polynomial. Two polynomials are equal if all their coefficients coincide. The polynomials with ${}a_{i}=0$ for all ${}i\geq 1$ are called constant polynomials, we write simply ${}a_{0}$ for them. In the zero polynomial, all coefficients equal ${}0$. Using the sum symbol, one can write a polynomial briefly as ${}\sum _{i=0}^{n}a_{i}X^{i}$.

Definition

The degree of a nonzero polynomial

${}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,$

with ${}a_{n}\neq 0$

is ${}n$.

The zero polynomial does not have a degree. The coefficient ${}a_{n}$, where ${}n$ is the degree of the polynomial, is called the leading coefficient of the polynomial. The term ${}a_{n}X^{n}$ is called the leading term of the polynomial.

The set of all polynomials over a field ${}K$ is called polynomial ring over ${}K$, it is denoted by ${}K[X]$, where ${}X$ is the variable of the polynomial ring.

Two polynomials

$P=\sum _{i=0}^{n}a_{i}X^{i}\,\,{\text{ and }}\,\,Q=\sum _{i=0}^{m}b_{i}X^{i}$

can be added by adding the components, i.e. the coefficients of the sum ${}P+Q$ are just the sums of the coefficients of the two polynomials. In case ${}n>m$, the "missing“ coefficients of ${}Q$ are to be interpreted as ${}0$. This addition is obviously associative and commutative, the zero polynomial is the neutral element and the negative polynomial ${}-P$ is obtained by taking from every coefficient of ${}P$ its negative.

One can also multiply two polynomials, one puts

${}X^{n}\cdot X^{m}:=X^{n+m}\,$

and extends this multiplication rule so that distributivity holds. This means one multiplies each summand with each summand and adds then everything together. One can describe this multiplication also explicitly by the rule:

${\left(\sum _{i=0}^{n}a_{i}X^{i}\right)}\cdot {\left(\sum _{j=0}^{m}b_{j}X^{j}\right)}=\sum _{k=0}^{n+m}c_{k}X^{k}{\text{ with }}c_{k}=\sum _{r=0}^{k}a_{r}b_{k-r}.$

For the degree the following rules hold.

1. ${}\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,.$

2. ${}\operatorname {deg} \,(P\cdot Q)=\operatorname {deg} \,(P)+\operatorname {deg} \,(Q)\,.$

The graph of a polynomial function from ${}\mathbb {R}$ to ${}\mathbb {R}$ of degree ${}5$.

If a polynomial ${}P\in K[X]$ and an element ${}a\in K$ are given, then one can insert ${}a$ into ${}P$ meaning that one replaces everywhere the variable ${}X$ by ${}a$. This yields a mapping

$K\longrightarrow K,a\longmapsto P(a),$

which is called the corresponding polynomial function.

If ${}P$ and ${}Q$ are polynomials, then the composition ${}P\circ Q$ is described in the following way: one has to replace everywhere in ${}P$ the variable ${}X$ by ${}Q$ and then simplify this expression. The result is again a polynomial. The order of ${}P$ and ${}Q$ is important for this.