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 .