**
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.*