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

Relating roots and coefficients of quadratics

When the roots of a polynomial are known, we can multiply out factors of the form (zrk) 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 az2 + bz + c, then the factored form will be given by a(zr)(zs) 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 rs = c/a.

A similar approach can be pursued for a polynomial of any degree. In general, for a polynomial of degree n, say an zn + an − 1 zn − 1 + ··· + a1 z + a0 , the sum of the roots is equal to an − 1/an, and the product of the roots is given by (−1)na0 /an. 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.