A polynomial is an expression of one variable of the form a_{n}x^{n} + a_{n-1}x^{n-1} + ^{ ... } + a_{2}x^{2} + a_{1}x + a_{0}, where a_{n}, a_{n-1},…, a_{1}, a_{0} are real numbers, n is a positive integer, and a_{n}≠ 0. The
degree of a polynomial is n. a_{n}, a_{n-1},…, a_{1}, a_{0} are the
coefficients of the polynomial. a_{n} is the leading coefficient, and a_{0}
is the constant term. a_{n}x^{n}, a_{n-1}x^{n-1},…, a_{2}x^{2}, a_{1}x, a_{0}
are the terms of the polynomial. There are n + 1 terms in a polynomial of
degree n.
A polynomial function is any function which is a polynomial; that is, it is
of the form f (x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ^{ ... } + a_{2}x^{2} + a_{1}x + a_{0}.
The roots of a polynomial function are the values of x for which the
function equals zero. Roots are also known as
zeros, x-intercepts, and solutions. All of these terms are synonymous. One
of the most important things to learn about polynomials is how to find their
roots.
Polynomial functions have special names depending on their degree. A polynomial
function of degree zero has only a constant term -- no x term. If the
constant is zero, that is, if the polynomial f (x) = 0, it is called the zero
polynomial. If the constant is not zero, then f (x) = a_{0}, and the
polynomial function is called a constant function. If the polynomial
function has degree one, then it is of the form f (x) = ax + b, and is called a
linear function. If the polynomial is of degree two, then it is of the form
f (x) = ax^{2} + bx + c, and is called a quadratic function. In the next
section, we'll learn more about quadratic functions.