## Polynomial function

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

For example we can use it to factor

Also frequently used are the following product formulas:

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

 (6)

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

and the remainder . Then the polynomial  can be expressed as

 (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

including possible multiplicities (that is, it could be , for example). Then (6) can be expressed as

where is the leading coefficient of the polynomial.

Department of Mathematics