Polynomial function

Polynomials. We define the $n$-th power $x^n$ of $x$ by $ {x^n = \underbrace{x\times x\times\cdots \times x}_{\mbox{$n$ times}}}$. A polynomial is of the form $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$. Each $a_k x^k$ of the polynomial is a term, and $a_k$ of the term is called a coefficient. In particular, the coefficient $a_n$ is called the leading coefficient of the polynomial if $a_n \neq 0$.

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 $(x - a)$. Occasionally you may be able to use the factoring formula

$\displaystyle x^2 + (a + b) x + ab = (x + a)(x + b).

For example we can use it to factor

$\displaystyle x^2 + x - 6 = x^2 + (3 + (-2))x + (3)(-2)
= (x + 3)(x - 2).

Also frequently used are the following product formulas:

(a \pm b)^2 = a^2 \pm 2ab + b^2
...ce{0.6in} (a \pm b)(a^2 \mp ab + b^2) = a^3 \pm b^3

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

$\displaystyle f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ (6)

with leading coefficient  $a_n \neq 0$.

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

$\displaystyle q(x) = a_n x^{n-1} + b_{1} x^{n-2} + \cdots + b_{n-2} x + b_{n-1},

and the remainder $r$. Then the polynomial $f(x)$ can be expressed as

$\displaystyle f(x) = (x - c) q(x) + r.$ (7)

Remainder Theorem. The formula (7) implies that we obtain $f(c) = r$ as the result of synthetic division. In particular, $f(c) = 0$ if and only if $(x - c)$ is a factor of $f(x)$, which we call factor theorem.

Zeros of polynomials. The zeros of a polynomial $f(x)$ are the solutions of the equation $f(x) = 0$. In particular, the polynomial (6) of degree $n$ has exactly $n$ zeros

$\displaystyle \gamma_1,\gamma_2,\ldots,\gamma_n,

including possible multiplicities (that is, it could be ${\gamma_1 = \gamma_2}$, for example). Then (6) can be expressed as

$\displaystyle f(x) = a_n(x - \gamma_1)(x - \gamma_2)\cdots(x - \gamma_n)

where $a_n$ is the leading coefficient of the polynomial.

Department of Mathematics
Last modified: 2005-09-29