The Journal of Online Mathematics and Its Applications, Volume 8 (2008)
The Most Marvelous Theorem in Mathematics, Dan Kalman

Terminology: coefficients, terms, factors, degree, roots

A polynomial is made up of powers of a variable multiplied by constants. The simplest type of polynomial is called a monomial, which is a constant multiplied by a whole number power of a variable. 3 z8, −1.2 z3, π z1, and 11/7 (considered to be multiplied by z0 = 1) are all monomials. In each case, the exponent is called the degree of the monomial, and the constant multiplier is called the coefficient. The constant 0 can be considered to be a monomial using any power of z, since 0zn = 0 for any n. For this reason, we only consider nonzero monomials to have a defined degree. The monomial 3z8 has degree 8 and a coefficient of 3. Throughout the webpages making up this article, we will allow coefficients to be any complex numbers, and we will think of z as standing for an unknown complex number.

A polynomial is a sum of a finite number of monomials, which are called the terms of the polynomial. The highest of the degrees of the terms is the degree of the polynomial. The coefficients of the terms are the coefficients of the polynomial. For example 4z3 − 2z2 + 7z − 1 is a polynomial of degree 3; the numbers 4, −2, 7, and −1 are the coefficients of this polynomial.

In algebra, it is understood that two expressions are equivalent if they produce identical results for any assignment of values to variables. When we write z2 − 1 = (z + 1)(z − 1), we know that the two sides of the equation will be equal for any choice of the variable z. Since z2 − 1 is a sum of monomials, it is a polynomial. On the other hand, (z + 1)(z − 1) is not a sum of monomials. Nevertheless, we consider it to be a polynomial. In general, any expression that is equivalent to a sum of monomials is a polynomial.

Any polynomial can be written in descending order. The general form is an zn + an −1 zn − 1 + ··· + a1z + a0 where n is a whole number and an is not zero. In this case, n is the degree of the polynomial.

Adding, subtracting, or multiplying two polynomials always produces another polynomial. If f(z), g(z), and p(z) are polynomials, and if f(z)g(z) = p(z), we say that f(z) and g(z) are factors of p(z). Thus, (z + 1) and (z − 1) are factors of z2 − 1.

The roots of a polynomial p(z) are the solutions to the equation p(z) = 0. The roots of z2 − 1 are the numbers 1 and −1. An important result from algebra states that any polynomial of degree n > 0 with coefficients that are complex numbers can be expressed in the form an(zr1)(zr2) ··· (zrn). The complex numbers r1, r2, ... , rn are the roots of this polynomial.