Polynomials

Definition

A single variable polynomial with real coefficients is a function $f$ that takes a real number as input and produces a real number as output and has the form

f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n

where $a_i$ are real numbers. The $a_i$ are called coefficients of $f$ . The degree of the polynomial is the integer $n$ .

Syntax:

a polynomial with real coefficients (the function $f$ ).
coefficients (the numbers $a_i$ ).
a polynomial's degree (the integer $n$ ).

Example:

g(t) = 2 + 0t + 4t^2 + (-1)t^3

Which can be written as

g(t) = 2 + 4t^2 - t^3

And can be evaluated for $t = 3$

g(3) = 2 + 4(3^2) - 3^3 = 11

Interesting ways to think about polynomials:

a polynomial as with any function can be represented as a set of pairs called points. That is if you take each input $t$ and pair it with its output $f(t)$ you get a set of tuples $(t, f(t))$ , which can be analysed from the perspective of set theory.
a polynomial's graph can be plotted as a curve in space, so that the horizontal direction represents the input and the vertical the output.
using the curves they "carve out" we can regard a polynomial as a geometric object, with properties like "curvature" and "smoothness".
any logical condition can be expressed as a combination of polynomials if we restrict their inpot to $1$ or $0$ :

AND(x,y) = xy \\ NOT(x) = 1 - x \\ OR(x,y) = 1 - (1-x)(1-y)