Polynomials

Overview

Let $R$ be a rng. A polynomial $f(x)$ with coefficients in $R$ is an infinite formal sum

where $a_i\in R$ and $a_i=0$ for all but a finite number of values of $i$. The $a_i$ are called the coefficients of $f(x)$. If for some $i\ge 0$ it is true that $a_i\ne 0$, the largest such value of $i$ is the degree of $f(x)$, denoted as $\text{deg}f$. If all $a_i=0$, then the degree of $f(x)$ is $-\mathrm{\infty }$.

We denote the set of all polynomials in an indeterminate $x$ with coefficients in $R$ as $R[x]$. Sometimes this set is denoted as $\mathcal{P}(R)$. The set of all polynomials with degree at most $m$ is then denoted as ${\mathcal{P}}_m(R)$. Such a set includes $0$.

Quadratics

A quadratic function (of a single variable) is a function of the form

where $a$, $b$, and $c$ are coefficients. The above form is called the standard form.

• If $a>0$, $f(x)$ is concave upward.
• If $a<0$, $f(x)$ is concave downward.
• The axis of symmetry is $x=-b/2a$.

A quadratic may have zero, one, or two real roots. It will always have exactly two complex roots. In both cases, these refer to the points $x$ where $f(x)=0$. If a quadratic has no real roots, we say it is irreducible.

Completing the Square

Completing the square is a technique for converting a quadratic polynomial in standard form to that of vertex form $a(x-h{)}^2+k$, where $h$ and $k$ are the $x$ and $y$ coordinates of the vertex respectively.

Quadratic Formula

Let $f(x)=a{x}^2+bx+c$. The quadratic formula is used to find the roots of $f$. It is given by

The portion under the radical is called the discriminant. If

• the discriminant is positive, $f$ has two real roots;
• the discriminant is negative, $f$ has two complex roots;
• the discriminant is zero, $f$ has one real repeated root.

Long Division

Polynomial long division is an algorithm for dividing a polynomial by another with equal or lower degree.

We say a rational function is proper if the degree of the numberator is less than that of the denominator. Otherwise we say it is improper. An improper rational function can be expressed as the sum of a polynomial and a proper rational function using long division.

Algebraic Numbers

The set of polynomials with coefficients in $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ are denoted by $\mathbb{Z}[x]$, $\mathbb{Q}[x]$, $\mathbb{R}[x]$, and $\mathbb{C}[x]$ respectively.

A number $x\in \mathbb{R}$ is called an algebraic number if it satisfies $p(x)=0$, where $p$ is a non-zero polynomial in $\mathbb{Z}[x]$. Otherwise it is called a transcendental number.