The Journal of Online Mathematics and Its Applications, Volume 8 (2008)

The Most Marvelous Theorem in Mathematics, Dan Kalman

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 `z`^{8}, −1.2 `z`^{3}, π `z`^{1}, and 11/7 (considered to be multiplied by `z`^{0} = 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 0`z`^{n} = 0 for any `n`. For this reason, we only consider nonzero monomials to have a defined degree. The monomial 3`z`^{8} 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 4`z`^{3} − 2`z`^{2} + 7`z ` − 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 `z`^{2} − 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 `z`^{2} − 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 `a`_{n} `z`^{n} + `a`_{n −1} `z`^{n − 1} + ··· + `a`_{1}`z` + `a`_{0} where `n` is a whole number and `a`_{n} 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 `z`^{2} − 1.

The *roots* of a polynomial `p`(`z`) are the solutions to the equation `p`(`z`) = 0. The roots of `z`^{2} − 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 `a`_{n}(`z` − `r`_{1})(`z` − `r`_{2}) ··· (`z` − `r`_{n}). The complex numbers `r`_{1}, `r`_{2}, ... , `r`_{n} are the roots of this polynomial.