# Polynomials

Notes on polynomials

A single variable polynomial is an expression of the form $f(x) = a_0 x^0 + a_1 x^1 + a_2 x^2 + ... a_n x^n$ where $a_i$ are coefficients and $n$ is the degree of the polynomial. Polynomials encapsulate the full expression of addition and multiplication.

We can calculate the *unique minimum*
$n$
degree polynomial that goes through a set of
$n + 1$
points
$\{ (x_0, y_0), (x_1, y_1), ..., (x_{n + 1}, y_{n + 1}) \}$
as
$f(x) = \sum_{i=0}^n y_i (\prod_{j \neq i} \frac{x - x_j}{x_i - x_j})$.

## Roots

Given a single variable polynomial $f(x)$, the roots of $f$ are the values $y$ such that $f(y) = 0$. Notice that its impossible for a polynomial to have more roots than its degree.

Given polynomials with real coefficients, we say that a
number
$x$
is *algebraic* (otherwise *trascendental*) if it
is the root of a polynomial whose coefficients are rational
numbers. Consider
$\sqrt{2}$
and polynomial
$f(x) = 1 - \frac{1}{2} x^2$.
Then
$f(\sqrt{2}) = 0$,
so
$\sqrt{2}$
is algebraic.

## Special Polynomials

### Zero Polynomial

The zero polynomial is defined as $f(x) = 0$ for any $x$, and it has a degree $-1$ by convention, as otherwise it would have every degree: $f(x) = 0x^0$, or $f(x) = 0x^0 + 0x^1$, or $f(x) = 0x^0 + 0x^1 + 0x^2$, etc.

Given polynomials $f$ and $g$, if $f - g$ is the zero polynomial, then it means that $f = g$.

## Operations

### Multiplication

Given an $n$ degree polynomial $f$ and an $m$ degree polynomial $g$, the polynomial $f \cdot g$ has a degree $n + m$. A polynomial multiplied by the zero polynomial is the zero polynomial.

## Properties

### Monic

A polynomial is *monic* if the non-zero coefficient of
highest degree (leading coefficient) is 1. For example, this
polynomial is monic:
$f(x) = 2 + 5x + x^2$,
as the highest degree part of the expression is
$x^2$,
and it has a coefficient 1, as
$x^2 = 1x^2$.