A polynomial is an expression of one variable of the form anxn + an-1xn-1 + ... + a2x2 + a1x + a0, where an, an-1, , a1, a0 are real numbers, n is a positive integer, and an≠ 0. The degree of a polynomial is n. an, an-1, , a1, a0 are the coefficients of the polynomial. an is the leading coefficient, and a0 is the constant term. anxn, an-1xn-1, , a2x2, a1x, a0 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) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0. 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) = a0, 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) = ax2 + bx + c, and is called a quadratic function. In the next section, we'll learn more about quadratic functions.