
Conjugate Zeros Theorem
If P(x) is a polynomial with real coefficients, and if a + bi is a zero of P, then a  bi is a zero of P.

Factor Theorem
If P(x) is a polynomial and P(a) = 0, then x  a is a factor of P(x). In other words, if the remainder when P(x) is divided by x  a is 0, then x  a is a factor of P(x).

Fundamental Theorem of Algebra
Every polynomial function of positive degree with complex coefficients has at least one complex zero.
Corollary. Every polynomial function of positive degree n has exactly n complex zeros (counting multiplicities). 
Multiplicity
A function with n identical roots is said to have a zero of multiplicity n.

Nested Form
The form of a polynomial P(x) = (((((a)x + b)x + c)x + d )x + ^{ ... }).

Rational Zeros Theorem
If P(x) is a polynomial with integer coefficients and if is a zero of P(x) (if P() = 0), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x).

Remainder Theorem
When a polynomial P(x) is divided by x  a, the remainder is equal to P(a).

Root
A number that, when plugged in for the variable, sets a function equal to zero. Also called a zero.

Synthetic Division
A process by which a polynomial is divided by a binomial, in which the coefficients of the polynomial are placed in a row and multiplied by and added to the constant divisor as in nested form.

Zero
A number that, when plugged in for the variable, sets a function equal to zero. Also called a root.