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

The Most Marvelous Theorem in Mathematics, Dan Kalman

When the roots of a polynomial are known, we can multiply out factors of the form (`z` − `r`_{k}) to obtain the coefficients of the polynomial. This leads to certain relationships between the roots and coefficients. Consider the particular case of a quadratic polynomial, that is, one of degree 2. If the original quadratic is `a``z`^{2} + `b``z` + `c`, then the factored form will be given by `a`(`z` − `r`)(`z` − `s`) where `r` and `s` are the roots. By equating these two different forms for the polynomial, we are led to

.

Now multiplying out the two factors on the left produces

This shows that −(`r` + `s`) = `b`/`a` and `r``s` = `c`/`a`.

A similar approach can be pursued for a polynomial of any degree. In general, for a polynomial of degree `n`, say `a`_{n} `z`^{n} + `a`_{n − 1} `z`^{n − 1} + ··· + `a`_{1} `z` + `a`_{0} , the sum of the roots is equal to −`a`_{n − 1}/`a`_{n}, and the product of the roots is given by (−1)^{n}`a`_{0} /`a`_{n}. There are also formulas for the other coefficients, but we will not need them for this article. For further information, see the article on elementary symmetric polynomials in Wikipedia.