Polynomials

A single variable polynomial is an expression of the form where are coefficients and is the degree of the polynomial. Polynomials encapsulate the full expression of addition and multiplication.

We can calculate the unique minimum degree polynomial that goes through a set of points as .

Roots

Given a single variable polynomial , the roots of are the values such that . Notice that its impossible for a polynomial to have more roots than its degree.

Given polynomials with real coefficients, we say that a number is algebraic (otherwise trascendental) if it is the root of a polynomial whose coefficients are rational numbers. Consider and polynomial . Then , so is algebraic.

Special Polynomials

Zero Polynomial

The zero polynomial is defined as for any , and it has a degree by convention, as otherwise it would have every degree: , or , or , etc.

Given polynomials and , if is the zero polynomial, then it means that .

Operations

Multiplication

Given an degree polynomial and an degree polynomial , the polynomial has a degree . 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: , as the highest degree part of the expression is , and it has a coefficient 1, as .