def: polynomial - an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

def: Degree of a Polynomial - the largest exponent of the polynomial

def: Turning Point - A point at which a graph changes directions. This happens at a peak or valley.

prop: If $n$ is the degree of a polynomial, then the polynomial can have at most $n - 1$ turning points

def: x-intecepts / zeros of a polynomial - the point(s) where a graph touches the x axis

prop: If $r$ is a zero of the graph of a polynomial, then $x - r$ is a factor of that polynomial

def: Multiplicity - given a polynomial in factored form, Multiplicity is the power of each factored piece.

example:

construct a polynomial of degree three whose zeros are $-2,0,2$. We are given three zeros, so we can form three factors.

$$ y = (x - (-2))(x - 0)(x - 2) $$

If we use multiplication to expand this expression in the usual way, we get

$$ y = x^3 - 4x $$

$$ \{ \mathbf{x_i}, y_i \} \qquad where \qquad i = 1 ... L, \quad y_i \in \{-1,1\}, \quad \mathbf{x} \in \mathfrak{R}^D $$

  1. http://sites.csn.edu/ehutchinson/Notes/3_2NOTES124.pdf

wikis http://tutorial.math.lamar.edu/Classes/Alg/Factoring.aspx https://en.wikipedia.org/wiki/Polynomial_ring https://en.wikipedia.org/wiki/Irreducible_polynomial https://en.wikipedia.org/wiki/Factorization_of_polynomials https://en.wikipedia.org/wiki/Factor_theorem