**Polynomials.**
We define the -th power of by
.
A *polynomial* is of the form
.
Each of the polynomial is a *term*,
and of the term is called a *coefficient*.
In particular, the coefficient is called the
*leading coefficient* of the polynomial if
.

**Factoring polynomials.**
In many cases we find by the method of ``trial and error''
that a polynomial of interest can be expressed as a product of polynomials of
the form .
Occasionally you may be able to use the factoring formula

**Polynomial functions.**
A function is called a *polynomial function of degree *
if

(6) |

with leading coefficient .

**Synthetic division.**
Suppose that is the polynomial function (6),
and that is any real number.
By using the guidelines for synthetic division in page 234,
we can always obtain the *quotient*

(7) |

**Remainder Theorem.**
The formula (7) implies that
we obtain as the result of synthetic division.
In particular,
if and only if is a factor of ,
which we call *factor theorem*.

**Zeros of polynomials.**
The *zeros* of a polynomial are the solutions
of the equation .
In particular, the polynomial (6) of degree has exactly
zeros

Department of Mathematics